Embeddings just below the stable range: classification

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1 Introduction

Most of this page is intended not only for specialists in embeddings, but also for mathematician from other areas who want to apply or to learn the theory of embeddings.

Recall the Whitney-Wu Unknotting Theorem: if $N$${{Authors|Askopenkov}} == Introduction == <wikitex>; Most of this page is intended not only for specialists in embeddings, but also for mathematician from other areas who want to apply or to learn the theory of embeddings. Recall the [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting theorems|Whitney-Wu Unknotting Theorem]]: if $N$ is a connected manifold of dimension $n>1$, and $m \ge2n+1$, then every two embeddings $N \to\Rr^m$ are isotopic \cite[Theorem 3.2]{Skopenkov2016c}, \cite[Theorem 2.5]{Skopenkov2006}. In this page we summarize the situation for $m=2n\ge6$ and $N$ is a connected, as well as in some more general situations. For the classification of embeddings of some disconnected manifolds see \cite{Skopenkov2016h}. For a [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|general introduction to embeddings]] as well as the [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Notation and conventions|notation and conventions]] used on this page, we refer to \cite[$\SN$ is a connected manifold of dimension $n>1$$n>1$, and $m \ge2n+1$$m \ge2n+1$, then every two embeddings $N \to\Rr^m$$N \to\Rr^m$ are isotopic [Skopenkov2016c, Theorem 3.2], [Skopenkov2006, Theorem 2.5]. In this page we summarize the situation for $m=2n\ge6$$m=2n\ge6$ and $N$$N$ is a connected, as well as in some more general situations. For the classification of embeddings of some disconnected manifolds see [Skopenkov2016h].

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$$\S$1, $\S$$\S$3]. Denote $1_n:=(1,0,\ldots,0)\in S^n$$1_n:=(1,0,\ldots,0)\in S^n$.

2 Classification

For the next theorem, the Whitney invariant $W$$W$ is defined in $\S$$\S$5 below.

Theorem 2.1. Assume that $N$$N$ is a closed connected $n$$n$-manifold, and either $n\ge4$$n\ge4$ or $n=3$$n=3$ and we are in the PL category.

(a) If $N$$N$ is oriented, the Whitney invariant,

$\displaystyle W:E^{2n}(N)\to H_1(N;\Zz_{\varepsilon(n-1)}),$ $$\displaystyle W:E^{2n}(N)\to H_1(N;\Zz_{\varepsilon(n-1)}),$$

is a 1-1 correspondence.

(b) If $N$$N$ is non-orientable, then there is a 1-1 correspondence

$\displaystyle E^{2n}(N)\to \begin{cases} H_1(N;\Zz_2) & n\text{ is odd}\\ \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s & n\text{ is even}\end{cases}.$ $$\displaystyle E^{2n}(N)\to \begin{cases} H_1(N;\Zz_2) & n\text{ is odd}\\ \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s & n\text{ is even}\end{cases}.$$

Remark 2.2 (Comments on the proof). Part (a) is proved in [Haefliger&Hirsch1963, Theorem 2.4] in the smooth category, and in [Weber1967, Theorem 4' in $\S$$\S$2], [Hudson1969, $\S$$\S$11], [Vrabec1977, Theorems 1.1 and 1.2] in the PL category, see also [Haefliger1962b, 1.3.e], [Haefliger1963], [Bausum1975, Theorem 43].

Part (b) is proved in [Bausum1975, Theorem 43] in the smooth category. By [Weber1967, Theorems 1 and 1' in $\S$$\S$2], [Skopenkov1997, Theorem 1.1.c] the proof works also in the PL category.

In Part (b) we replaced the kernel $\ker Sq^1$$\ker Sq^1$ from [Bausum1975, Theorem 43] by $\Zz_2^{s-1}$$\Zz_2^{s-1}$. This is possible because, as a specialist could see, $Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$$Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$ is given by multiplication with the first Stefel-Whitney class $w_1(N)$$w_1(N)$ (which equals to the first Wu class $v_1(N)$$v_1(N)$ [Milnor&Stasheff1974, Theorem 11.4]). Since $N$$N$ is non-orientable, $w_1(N)\neq 0$$w_1(N)\neq 0$. So by Poincaré duality, $\ker Sq^1 \cong \Zz_2^{s-1}$$\ker Sq^1 \cong \Zz_2^{s-1}$.

The 1-1 correspondence from (b) can presumably be defined as a (generalized) Whitney invariant, see [Vrabec1977], but the proof used 'the Haefliger-Wu invariant' whose definition can be found e.g. in [Skopenkov2006, $\S$$\S$5]. It would be interesting to check if part (b) is equivalent to different forms of description of $E^{2n}(N)$$E^{2n}(N)$ [Haefliger1962b, 1.3.e], [Haefliger1963], [Vrabec1977, Theorem 1.1].

The classification of smooth embeddings of 3-manifolds in $\Rr^6$$\Rr^6$ is more complicated, see Theorem 6.3 below for $l=1$$l=1$ or [Skopenkov2016t].

Concerning embeddings of connected $n$$n$-manifolds in $\Rr^{2n-1}$$\Rr^{2n-1}$ see [Yasui1984] for $n\ge5$$n\ge5$, [Skopenkov2016f] for $n=4$$n=4$, and [Saeki1999], [Skopenkov2010], [Tonkonog2010] for manifolds with boundary.

Theorem 2.1 is generalized to a description of $E^{2n-k}(N)$$E^{2n-k}(N)$ for closed $k$$k$-connected $n$$n$-manifolds $N$$N$, see Theorem 6.2.

3 Hudson and Lagrangian tori

Together with the Haefliger knotted sphere [Skopenkov2016t, Example 2.1], [Skopenkov2006, Example 3.4], the examples of Hudson tori presented below were the first examples of non-isotopic embeddings in codimension greater than 2. (Hudson's construction [Hudson1963] was not as explicit as those below.) Abbreviate

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${\rm i}_{2n,n-1}$ to just

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${\rm i}$.

Example 3.1. Let us construct, for any $a\in\Zz$$a\in\Zz$ and $n\ge2$$n\ge2$, a smooth embedding

$\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.$ $$\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.$$

We start with the cases $a=0,1$$a=0,1$.

Take the standard inclusion $\partial D^2\subset\partial D^{n+1}$$\partial D^2\subset\partial D^{n+1}$. The 'standard embedding' $\Hud_n(0)$$\Hud_n(0)$ is given by the standard inclusions

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$$\displaystyle \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\overset{{\rm i}}\to\Rr^{2n}.$$

Define the `standard embedding'

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$2{\rm i}:2D^{n+1}\times S^{n-1}\to\Rr^{2n}$ analogously to

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${\rm i}$, where $2$$2$ means homothety with coefficient 2.

Take the embedding $g_1$$g_1$ given by

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$$\displaystyle S^n=2\partial D^{n+1}\times1_{n-1}\subset 2D^{n+1}\times S^{n-1}\overset{2{\rm i}}\to\Rr^{2n}.$$

The segment

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$2{\rm i}([1,2]1_n\times1_{n-1})$ joins the images of $\Hud_n(0)$$\Hud_n(0)$ and $g_1$$g_1$; the interior of this segment misses the images. Let $\Hud_n(1)$$\Hud_n(1)$ be the linked embedded connected sum of $\Hud_n(0)$$\Hud_n(0)$ and $g_1$$g_1$ along this segment, compatible with the orientation, cf. [Avvakumov2017, $\S$$\S$1.5]. (Here 'linked' means that the images of the embeddings are not contained in disjoint cubes, unlike for the unlinked embedded connected sum [Skopenkov2016c, $\S$$\S$5].)

For $a\in\Zz$$a\in\Zz$ we repeat the above construction of $g_1$$g_1$ replacing $2\partial D^{n+1}\times1_{n-1}$$2\partial D^{n+1}\times1_{n-1}$ by $|a|$$|a|$ copies $(1+\frac1k)\partial D^{n+1}\times1_{n-1}$$(1+\frac1k)\partial D^{n+1}\times1_{n-1}$ of $S^n$$S^n$, $k=1,\ldots,|a|$$k=1,\ldots,|a|$. The copies are outside $D^{n+1}\times S^{n-1}$$D^{n+1}\times S^{n-1}$ and are `parallel' to $\partial D^{n+1}\times1_{n-1}$$\partial D^{n+1}\times1_{n-1}$. The copies have the standard orientation for $a>0$$a>0$ or the opposite orientation for $a<0$$a<0$. Then we make embedded connected sum along natural segments joining every $k$$k$-th copy to the $(k+1)$$(k+1)$-th copy. We obtain an embedding $g_a:S^n\to\Rr^{2n}$$g_a:S^n\to\Rr^{2n}$ which has disjoint images with $\Hud_n(0)$$\Hud_n(0)$. Let $\Hud_n(a)$$\Hud_n(a)$ be the linked embedded connected sum of $\Hud_n(0)$$\Hud_n(0)$ and $g_a$$g_a$.

The original motivation for Hudson was that $\Hud_n(1)$$\Hud_n(1)$ is not isotopic to $\Hud_n(0)$$\Hud_n(0)$ for any $n\ge3$$n\ge3$ (this is a particular case of Proposition 3.2 below). One might guess that $\Hud_n(a)$$\Hud_n(a)$ is not isotopic to $\Hud_n(a')$$\Hud_n(a')$ for $a\ne a'$$a\ne a'$ and that a $\Zz$$\Zz$-valued invariant of $E^{2n}(S^1 \times S^{n-1})$$E^{2n}(S^1 \times S^{n-1})$ can be defined by the homotopy class of the map

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$$\displaystyle S^n\overset g\to S^{2n}-{\rm i}(D^{n+1}\times S^{n-1})\sim S^{2n}-S^{n-1}\sim S^n \quad\text{which is}\quad a\in\pi_n(S^n)\cong\Zz.$$

However, this is only true for $n$$n$ odd.

Proposition 3.2. For $n\ge3$$n\ge3$ odd $\Hud_n(a)$$\Hud_n(a)$ is isotopic to $\Hud_n(a')$$\Hud_n(a')$ if and only if $a=a'$$a=a'$.

For $n\ge4$$n\ge4$ even $\Hud_n(a)$$\Hud_n(a)$ is isotopic to $\Hud_n(a')$$\Hud_n(a')$ if and only if $a\equiv a'\text{ mod}2$$a\equiv a'\text{ mod}2$.

Proposition 3.2 follows by calculation of the Whitney invariant (Remark 5.3.d below) and, for $n$$n$ even, by Theorem 2.1. This proposition holds with the same proof in the piecewise smooth category, whose definition is recalled in [Skopenkov2016f, Remark 1.1]). Proposition 3.2 also holds in the PL category (with an analogous construction of $\Hud_n(a)$$\Hud_n(a)$ for the PL category). It would be interesting to find an explicit construction of an isotopy between $\Hud_{2k}(a)$$\Hud_{2k}(a)$ and $\Hud_{2k}(a+2)$$\Hud_{2k}(a+2)$, cf. [Vrabec1977, $\S$$\S$5]. Analogously, $\Hud_2(a)$$\Hud_2(a)$ is not isotopic to $\Hud_2(a')$$\Hud_2(a')$ if $a\not\equiv a'\text{ mod}2$$a\not\equiv a'\text{ mod}2$. It would be interesting to know if the converse holds, e.g. is $\Hud_2(0)$$\Hud_2(0)$ (PS or smoothly) isotopic to $\Hud_2(2)$$\Hud_2(2)$.

Example 3.3. Take any $a\in\Zz$$a\in\Zz$. Take a map $\overline a:S^{n-1}\to S^{n-1}$$\overline a:S^{n-1}\to S^{n-1}$ of degree $a$$a$ (so we can take $\overline 1=\id$$\overline 1=\id$). Recall that $D^{n+1}=\{(y,x)\in D^n\times D^1\ :\ |y|^2+|x|^2\le1\}$$D^{n+1}=\{(y,x)\in D^n\times D^1\ :\ |y|^2+|x|^2\le1\}$. Define the smooth embedding $\Hud_n'(a)$$\Hud_n'(a)$ to be the composition

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$$\displaystyle S^1\times S^{n-1}\overset{\widehat a}\to D^{n+1}\times S^{n-1}\overset{\rm i}\to\Rr^{2n} \quad\text{where}\quad \widehat a(e^{i\theta},y):=(\overline a(y)\cos\theta,\sin\theta,y).$$

Figure 1: The Hudson torus

Let us present a geometric description of this embedding. Define a map $\widetilde a:S^0\times S^{n-1}\to S^{n-1}$$\widetilde a:S^0\times S^{n-1}\to S^{n-1}$ by $\widetilde a(s,t):=s\overline a(t)$$\widetilde a(s,t):=s\overline a(t)$. This map gives an embedding

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$$\displaystyle S^0\times S^{n-1}\xrightarrow{\widetilde a\times {\rm pr}_2}S^{n-1}\times S^{n-1}\subset D^{n+1}\times S^{n-1}\overset{{\rm i}}\to\Rr^{2n}.$$

See Figure 1. The image of

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$\widetilde a\times{\rm pr}_2$ is the union of the graphs of the maps $\overline a$$\overline a$ and $-\overline a$$-\overline a$. For any $t\in S^{n-1}$$t\in S^{n-1}$ the disk

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${\rm i}(D^{n+1}\times t)$ intersects the image at two points lying in

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${\rm i}(D^n\times t)$, i.e., at the image of an embedding

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$S^0\times t\to {\rm i}(D^n\times t)$. The embedding $\Hud_n'(a)$$\Hud_n'(a)$ is obtained by extending the latter embeddings to embeddings

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$S^1\times t\to {\rm i}(D^{n+1}\times t)$ for all $t$$t$. See Figure 2.

Figure 2: To the construction of the Hudson torus

Remark 3.4. (a) The analogue of Proposition 3.2 for $\Hud_n$$\Hud_n$ replaced to $\Hud_n'$$\Hud_n'$ holds, with an analogous proof.

(b) The embeddings $\Hud_n(a)$$\Hud_n(a)$ and $\Hud_n'(a)$$\Hud_n'(a)$ are smoothly isotopic for $n\ge4$$n\ge4$ and are PS isotopic for $n\ge3$$n\ge3$ [Skopenkov2006a, commutativity of the left upper square in the Restriction Lemma 5.2], [Skopenkov2015a, Lemma 2.15.c] (see [Skopenkov2016f, Remark 1.2]). This follows by calculation of the Whitney invariant (Remark 5.3.d below). It would be interesting to know if they are smoothly isotopic for $n=3$$n=3$. It would be interesting to know if they are piecewise smoothly isotopic for $n=2$$n=2$.

(b') Although construction of $\Hud_n(a)$$\Hud_n(a)$ and $\Hud_n'(a)$$\Hud_n'(a)$ is explicit, no explicit construction of an isotopy between them is known. Proof from (b) of their being isotopic is by calculating invariants, and using completeness result for the invariants. This proof is fairly constructive, so in principle it does give a specific isotopy. However, the construction is so complicated that one would not call such an isotopy explicit. This is analogous to the Smale `turning sphere inside out' theorem (although the Smale theorem concerns regular homotopy of immersions not isotopy of embeddings).

In higher dimensions, a proof that any embedding is isotopic to some of the constructed embeddings is even less likely to use an explicit construction. The only proofs we have go through calculating the invariants and using classification results.

(c) For $n=2$$n=2$ Example 3.3 gives what we call the 'left' Hudson torus. The 'right' Hudson torus is constructed analogously. It is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of $S^1\times S^1$$S^1\times S^1$. The right and the left Hudson tori are not isotopic by Remark 5.3.d below.

(d) Analogously one constructs the Hudson torus $\Hud_{n,p}(a):S^p\times S^{n-p}\to\Rr^{2n-p+1}$$\Hud_{n,p}(a):S^p\times S^{n-p}\to\Rr^{2n-p+1}$ for $a\in\Zz$$a\in\Zz$ and $n>p\ge0$$n>p\ge0$ or, more generally, $\Hud_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$$\Hud_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$ for $a\in\pi_n(S^{m-n+p-1})$$a\in\pi_n(S^{m-n+p-1})$ and $m>n>p\ge0$$m>n>p\ge0$. There are versions $\Hud'_{m,n,p}(a)$$\Hud'_{m,n,p}(a)$ of these constructions corresponding to Definition 3.3. For $p=0$$p=0$ this corresponds to the Zeeman map [Skopenkov2016h, Definition 2.2] and its composition with 'the unframed second Kirby move' [Skopenkov2015a, $\S$$\S$2.3]. It would be interesting to know if the links $\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$$\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$ are isotopic, cf. [Skopenkov2015a, Remark 2.7.b]. These constructions can be further generalized [Skopenkov2016k].

Example 3.5. Define a smooth embedding

$\displaystyle l:S^1\times S^{n-1}\to \C^n=\R^{2n}\quad\text{by}\quad l(z,x) := \left(1+\frac z2\right)x.$ $$\displaystyle l:S^1\times S^{n-1}\to \C^n=\R^{2n}\quad\text{by}\quad l(z,x) := \left(1+\frac z2\right)x.$$

Here $S^1\subset\R^2=\C$$S^1\subset\R^2=\C$ and $S^{n-1}\subset\R^n\subset\C^n$$S^{n-1}\subset\R^n\subset\C^n$. This embedding is known to be Lagrangian; see more in [Nemirovski2024].

4 Action by linked embedded connected sum

In this section we generalize the construction of the Hudson torus $\Hud(a)$$\Hud(a)$. Let $N$$N$ be a closed connected oriented $n$$n$-manifold. We work in the smooth category which we omit. Apparently analogous results hold for $n\ge3$$n\ge3$ in the PL and PS categories (see [Skopenkov2016f, Remark 1.2]).

Example 4.1. For any $n\ge4$$n\ge4$, an embedding $f_0:N\to\Rr^{2n}$$f_0:N\to\Rr^{2n}$ and $a\in H_1(N;\Zz)$$a\in H_1(N;\Zz)$, we shall construct an embedding $f_a:N\to\Rr^{2n}$$f_a:N\to\Rr^{2n}$. This embedding is said to be obtained by linked embedded connected sum of $f_0$$f_0$ with an $n$$n$-sphere representing the `homology Alexander dual' $A:=\widehat{A_{f_0}}a\in H_n(C_{f_0})$$A:=\widehat{A_{f_0}}a\in H_n(C_{f_0})$ of $a$$a$ (defined in [Skopenkov2005, Alexander Duality Lemma 4.6]).

Represent $a$$a$ by an embedding $a:S^1\to N$$a:S^1\to N$. By definition, the class $A$$A$ is represented by properly oriented $\nu_{f_0}^{-1}a(S^1)$$\nu_{f_0}^{-1}a(S^1)$. Since any orientable bundle over $S^1$$S^1$ is trivial, $\nu_{f_0}^{-1}a(S^1)\cong S^1\times S^{n-1}$$\nu_{f_0}^{-1}a(S^1)\cong S^1\times S^{n-1}$. Take an embedding $g:S^1\times S^{n-1}\to C_{f_0}$$g:S^1\times S^{n-1}\to C_{f_0}$ whose image is $\nu_{f_0}^{-1}a(S^1)$$\nu_{f_0}^{-1}a(S^1)$ and which represents $A$$A$. By embedded surgery on $S^1\times1_{n-1}\subset S^1\times S^{n-1}$$S^1\times1_{n-1}\subset S^1\times S^{n-1}$ we obtain an embedding $g_1:S^n\to C_{f_0}$$g_1:S^n\to C_{f_0}$ representing $A$$A$ (see details in Proposition 4.2 below). Define $f_a$$f_a$ to be the linked embedded connected sum of $f_0$$f_0$ and $g_1$$g_1$, along some arc joining their images.

Proposition 4.2 (Embedded surgery). For any $n\ge3$$n\ge3$, a neighborhood $U$$U$ of a codimension at least 3 subpolyhedron in $\Rr^{2n}$$\Rr^{2n}$ and an embedding $g:S^1\times S^{n-1}\to\Rr^{2n}-U$$g:S^1\times S^{n-1}\to\Rr^{2n}-U$ there is an embedding $g_1:S^n\to\Rr^{2n}-U$$g_1:S^n\to\Rr^{2n}-U$ homologous to $g$$g$.

Proof. Take a vector field on $g(S^1\times1_{n-1})$$g(S^1\times1_{n-1})$ normal to $g(S^1\times S^{n-1})$$g(S^1\times S^{n-1})$. Extend $g|_{S^1\times1_{n-1}}$$g|_{S^1\times1_{n-1}}$ along this vector field to a map $\overline b:D^2\to\Rr^{2n}$$\overline b:D^2\to\Rr^{2n}$.

Since $2n>4$$2n>4$ and $U$$U$ is a neighborhood $U$$U$ of a codimension at least 3 subpolyhedron, by general position we may assume that $\overline b$$\overline b$ is an embedding and that

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$\overline b({\rm Int}D^2)$ misses $U\cup g(S^1\times S^{n-1})$$U\cup g(S^1\times S^{n-1})$.

Since $n-1>1$$n-1>1$, we have $\pi_1(V_{2n-2,n-1})=0$$\pi_1(V_{2n-2,n-1})=0$. Hence the standard $(n-1)$$(n-1)$-framing of $S^1\times1_{n-1}$$S^1\times1_{n-1}$ in $S^1\times S^{n-1}$$S^1\times S^{n-1}$ extends to an $(n-1)$$(n-1)$-framing on $\overline b(D^2)$$\overline b(D^2)$ in $\Rr^{2n}$$\Rr^{2n}$. Thus $\overline b$$\overline b$ extends to an embedding

$\displaystyle \widehat b:D^2\times D^{n-1}\to\R^{2n}-U\quad\text{such that}\quad \widehat b(\partial D^2\times D^{n-1})\subset g(S^1\times S^{n-1}).$ $$\displaystyle \widehat b:D^2\times D^{n-1}\to\R^{2n}-U\quad\text{such that}\quad \widehat b(\partial D^2\times D^{n-1})\subset g(S^1\times S^{n-1}).$$

Take an embedding $g_1:S^n\to\R^{2n}-U$$g_1:S^n\to\R^{2n}-U$ such that

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$$\displaystyle g_1(S^n)\ =\ g(S^1\times S^{n-1})-\widehat b(\partial D^2\times{\rm Int}D^{n-1}) \bigcup\limits_{\widehat b(\partial D^2\times\partial D^{n-1})} \widehat b(D^2\times\partial D^{n-1})\ \cong\ S^n.$$

with proper orientation so that $g_1$$g_1$ is homologous to $g$$g$. QED

The isotopy class of the embedding $f_a$$f_a$ is independent of the choises in the construction. The independence of the arc and of the maps $g,g_1$$g,g_1$ follows by $n\ge3$$n\ge3$ and by Proposition 4.3 below, respectively.

By Definition 5.1 of the Whitney invariant, $W(f_a,f_0)$$W(f_a,f_0)$ is $a$$a$ for $n\ge3$$n\ge3$ odd and $a \mod2$$a \mod2$ for $n\ge4$$n\ge4$ even. Thus by Theorem 2.1.a for $n\ge4$$n\ge4$ all isotopy classes of embeddings $N\to\Rr^{2n}$$N\to\Rr^{2n}$ can be obtained from any chosen embedding $f_0$$f_0$ by the above construction.

Proposition 4.3. For any $n\ge4$$n\ge4$ both the linked embedded connected sum and parametric connected sum (introduced in [Skopenkov2006a], [Skopenkov2015a]) define free transitive actions of $H_1(N;\Zz_{\varepsilon(n-1)})$$H_1(N;\Zz_{\varepsilon(n-1)})$ on $E^{2n}(N)$$E^{2n}(N)$.

This follows by Theorem 2.1.a and by [Skopenkov2014, Remark 18.a].

5 The Whitney invariant

Let $N$$N$ be a closed $n$$n$-manifold. Take an embedding $f_0:N\to\Rr^m$$f_0:N\to\Rr^m$. Fix an orientation on $\Rr^m$$\Rr^m$. For any other embedding $f \colon N \to \Rr^m$$f \colon N \to \Rr^m$ we define the Whitney invariant

$\displaystyle W(f, f_0)=W_{f_0}(f)=W(f)\in H_{2n-m+1}(N;\Zz_N).$ $$\displaystyle W(f, f_0)=W_{f_0}(f)=W(f)\in H_{2n-m+1}(N;\Zz_N).$$

Here the coefficients $\Zz_N$$\Zz_N$ are $\Zz$$\Zz$ if $N$$N$ is oriented and $m-n$$m-n$ is odd, and are $\Zz_2$$\Zz_2$ otherwise.

Roughly speaking,

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$W(f):=[{\rm Cl}\Sigma(H)]$ is defined as the homology class of the closure of the self-intersection set of a general position homotopy $H$$H$ between $f$$f$ and $f_0$$f_0$.

This is formalized in Definition 5.2 in the smooth category, following [Skopenkov2010], see also [Haefliger&Hirsch1963]. The definition in the PL category is analogous [Hudson1969, $\S$$\S$11], [Vrabec1977, p. 145], [Skopenkov2006, $\S$$\S$2.4 `The Whitney invariant']. We begin by presenting a simpler definition, Definition 5.1, for a particular case.

For Theorem 2.1 only the case $m=2n$$m=2n$ is required.

Definition 5.1. Assume that $N$$N$ is $(2n-m)$$(2n-m)$-connected and $2m\ge3n+3$$2m\ge3n+3$. Then by [Haefliger&Hirsch1963, Theorem 3.1.b] restrictions of $f$$f$ and $f_0$$f_0$ to $N_0$$N_0$ are isotopic, cf. [Takase2006, Lemma 2.2]. (Here is sketch of an argument. Using the Smale-Hirsch classification of immersions we obtain that restrictions of $f$$f$ and $f_0$$f_0$ to $N_0$$N_0$ are `regular homotopic', see [Koschorke2013, Definition 2.7]. Since $N$$N$ is $(2n-m)$$(2n-m)$-connected, $N_0$$N_0$ retracts to an $(m-n-1)$$(m-n-1)$-dimensional polyhedron. Therefore these restrictions are isotopic.)

So we can make an isotopy of $f$$f$ and assume that $f=f_0$$f=f_0$ on $N_0$$N_0$. Take a general position homotopy $F:B^n\times I\to\Rr^m$$F:B^n\times I\to\Rr^m$ relative to $\partial B^n$$\partial B^n$ between the restrictions of $f$$f$ and $f_0$$f_0$ to $B^n$$B^n$. Let $f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$$f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$ (`the intersection of this homotopy with $f(N-B^n)$$f(N-B^n)$').

Since $n+2(n+1)<2m$$n+2(n+1)<2m$, by general position

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${\rm Cl}(f\cap F)$ is a compact $(2n+1-m)$$(2n+1-m)$-manifold whose boundary is contained in $\partial N_0$$\partial N_0$.

So $f\cap F$$f\cap F$ carries a homology class with $\Zz_2$$\Zz_2$ coefficients. If $m-n$$m-n$ is odd and $N$$N$ is oriented, then $f\cap F$$f\cap F$ has a natural orientation defined below, and so carries a homology class with $\Zz$$\Zz$ coefficients. Define $W(f)$$W(f)$ to be the homology class:

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$$\displaystyle W(f):=[{\rm Cl}(f\cap F)]\in H_{2n-m+1}(N_0,\partial N_0;\Zz_N)\cong H_{2n-m+1}(N;\Zz_N).$$

The orientation on $f\cap F$$f\cap F$ (extendable to

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${\rm Cl}(f\cap F)$) is defined (for $m-n$$m-n$ odd and $N$$N$ is oriented) as follows (cf. Remark 7.3). For any point $x_f\in f\cap F$$x_f\in f\cap F$ take a base $\xi_f$$\xi_f$ at $x_f$$x_f$ tangent to $f\cap F$$f\cap F$. Complete this base to a positive base $(\xi_f,\eta_f)$$(\xi_f,\eta_f)$ tangent to $N$$N$. Since $n+2(n+1)<2m$$n+2(n+1)<2m$, by general position there is a unique point $x_F\in B^n\times I$$x_F\in B^n\times I$ such that $Fx_F=fx_f$$Fx_F=fx_f$. The tangent base $\xi_f$$\xi_f$ at $x_f$$x_f$ thus gives a base $\xi_F$$\xi_F$ at $x_F$$x_F$ tangent to $B^n\times I$$B^n\times I$ such that $df(x_f)\xi_f=dF(x_F)\xi_F$$df(x_f)\xi_f=dF(x_F)\xi_F$. Complete this base $\xi_F$$\xi_F$ to a positive base $(\xi_F,\eta_F)$$(\xi_F,\eta_F)$ tangent to $B^n\times I$$B^n\times I$, where the orientation on $B^n$$B^n$ comes from $N$$N$. The union $\zeta=(df(x_f)\xi_f,df(x_f)\eta_f,dF(x_F)\eta_F)$$\zeta=(df(x_f)\xi_f,df(x_f)\eta_f,dF(x_F)\eta_F)$ of the images of the constructed bases is a base at $fx_f=Fx_F$$fx_f=Fx_F$ of $\Rr^m$$\Rr^m$. If $\zeta$$\zeta$ is positive, then call the tangent base $\xi_f$$\xi_f$ of $f\cap F$$f\cap F$ `positive'. Since a change of the orientation on $f\cap F$$f\cap F$ forces a change of the orientation of $\zeta$$\zeta$, this condition indeed defines an orientation on $f\cap F$$f\cap F$.

Definition 5.2. Assume that $m\ge n+2$$m\ge n+2$. Take a general position homotopy $H:N\times I\to\Rr^m\times I$$H:N\times I\to\Rr^m\times I$ between $f_0$$f_0$ and $f$$f$.

The closure

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${\rm Cl}\Sigma(H)$ of the self-intersection set carries a cycle mod 2. If $N$$N$ is oriented and $m-n$$m-n$ is odd, the closure also carries an integer cycle. See [Hudson1967, $\S$$\S$11], [Skopenkov2006, $\S$$\S$2.3 `The Whitney obstruction']. Let us informally explain these facts. For $2m\ge3n+2$$2m\ge3n+2$ by general position the closure

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${\rm Cl}\Sigma(H)$ can be assumed to be a submanifold. In general, since $m\ge n+2$$m\ge n+2$, by general position the closure has codimension 2 singularities, as defined in $\S$$\S$7. So the closure carries a cycle mod 2. When $m-n$$m-n$ is odd the closure also has a canonical orientation (see Definition 7.1 and Remark 7.2), so the closure carries an integer cycle.

Define the Whitney invariant to be the homology class:

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$$\displaystyle W(f):=[{\rm Cl}\Sigma(H)]\in H_{2n-m+1}(N\times I;\Zz_N)\cong H_{2n-m+1}(N;\Zz_N).$$

Clearly, $W(f) = W(f')$$W(f) = W(f')$ if $f$$f$ is isotopic to $f'$$f'$. Hence the Whitney invariant defines a map

$\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_N),\quad [f] \mapsto W(f).$ $$\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_N),\quad [f] \mapsto W(f).$$

Clearly, $W(f_0)=0$$W(f_0)=0$ (for both definitions).

The definition of $W$$W$ depends on the choice of $f_0$$f_0$, but we write $W$$W$ not $W_{f_0}$$W_{f_0}$ for brevity.

Remark 5.3. (a) The Whitney invariant is well-defined by Definition 5.2, i.e. is independent of the choice of a general position homotopy $H:N\times I\to\Rr^m\times I$$H:N\times I\to\Rr^m\times I$ from $f_0$$f_0$ to $f$$f$.

This follows from the equality

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$[{\rm Cl}\Sigma(H_0)]−[{\rm Cl}\Sigma(H_1)] = \partial [{\rm Cl}\Sigma(H_{01})]$ for a general position homotopy $H_{01}:N\times I\times I\to\Rr^m\times I\times I$$H_{01}:N\times I\times I\to\Rr^m\times I\times I$ between general position homotopies $H_0,H_1:N\times I\to\Rr^m\times I$$H_0,H_1:N\times I\to\Rr^m\times I$ from $f_0$$f_0$ to $f$$f$. See details in [Hudson1969, $\S$$\S$11].

(b) Definition 5.1 is a particular case of Definition 5.2. (Indeed, if $f=f_0$$f=f_0$ on $N_0$$N_0$, we can take $H$$H$ to be fixed on $N_0$$N_0$. See details in [Skopenkov2010, Difference Lemma 2.4].) Hence the Whitney invariant is well-defined by Definition 5.1, i.e. independent of the choice of $F$$F$ and of the isotopy making $f=f_0$$f=f_0$ outside $B^n$$B^n$.

(c) The class $W(f)$$W(f)$ is independent of the choice of the orientation on $N$$N$ (because a change of the orientation on $N$$N$ forces a change of the orientation on $f\cap F$$f\cap F$ or on

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${\rm Cl}\Sigma(H)$). For the reflection $\sigma:\Rr^m\to\Rr^m$$\sigma:\Rr^m\to\Rr^m$ with respect to a hyperplane we have $W(\sigma\circ f)=-W(f)$$W(\sigma\circ f)=-W(f)$ (because a change of the orientation on $\Rr^m$$\Rr^m$ forces a change of the orientation on $f\cap F$$f\cap F$ or on

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${\rm Cl}\Sigma(H)$; for Definition 5.1 also observe that we may assume that $f=f_0=\sigma\circ f$$f=f_0=\sigma\circ f$ on $N_0$$N_0$).

(d) For the Hudson tori $W(\Hud_n(a))=W(\Hud'_n(a))$$W(\Hud_n(a))=W(\Hud'_n(a))$ is $a$$a$ or $a\mod2$$a\mod2$ for $n\ge3$$n\ge3$, and $W(\Hud'_2(a))=(a\mod2,0)$$W(\Hud'_2(a))=(a\mod2,0)$.

For $\Hud_n(a)$$\Hud_n(a)$ this is clear by Definition 5.1. For $\Hud'_n(a)$$\Hud'_n(a)$ and $n\ge3$$n\ge3$ this was proved in [Hudson1963] (with a different but equivalent definition of the Whitney invariant; using and proving a particular case of Remark 5.3.f). For $\Hud'_2(a)$$\Hud'_2(a)$ the proof is analogous.

(e) $W(f\#g)=W(f)$$W(f\#g)=W(f)$ for any pair of embeddings $f:N\to\Rr^m$$f:N\to\Rr^m$ and $g:S^n\to\Rr^m$$g:S^n\to\Rr^m$. This is clear by Definition 5.1 because $W(f\#g)-W(f)=W(f\#g,f_0)-W(f,f_0)=W(f\#g,f)=0$$W(f\#g)-W(f)=W(f\#g,f_0)-W(f,f_0)=W(f\#g,f)=0$. Let us prove the latter equality. Take the identical isotopy $H_f$$H_f$ of $f$$f$ and a general position homotopy $H_g$$H_g$ between $g$$g$ and the standard embedding. Then the boundary connected sum $H_f\sharp H_g$$H_f\sharp H_g$ is a general position homotopy between $f\#g$$f\#g$ and an embedding isotopic to $f$$f$. The cycle

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${\rm Cl}\Sigma(H_f\sharp H_g)={\rm Cl}\Sigma H_g$ is null-homologous in $S^n$$S^n$ and hence in $N\cong N\#S^n$$N\cong N\#S^n$; cf. [Skopenkov2008, Addendum to the Classification Theorem].

(f) For $m=2n+1$$m=2n+1$ and $N=S^n\sqcup S^n$$N=S^n\sqcup S^n$ the Whitney invariant equals to the pair of linking coefficients [Skopenkov2016h, $\S$$\S$3].

(g) The Whitney invariant need not be a bijection for $m<2n$$m<2n$. This is seen, for example, by applying Theorem 6.4 below in case of knotted tori [Skopenkov2016k, Theorem 5.1]) or by taking $n$$n$ even, $N$$N$ non-orientable, $m=2n$$m=2n$ and applying by Theorem 2.1.b.

(h) The paper [Haefliger&Hirsch1963, $\S$$\S$2] gives yet another equivalent definition of the Whitney invariant, and proves its injectivity under certain restrictions (the Haefliger-Hirsch $h$$h$-principle, see Theorem 6.2).

(i) Let $N$$N$ be a closed orientable $n$$n$-manifold. For two embeddings $f,g:N\to\R^{2n-1}$$f,g:N\to\R^{2n-1}$ the invariant defined in [Haefliger&Hirsch1963, $\S$$\S$2] equals plus or minus their Whitney invariant $W(f,g)$$W(f,g)$ from Definition 5.1. This is proved analogously to the following version.

Let $D^n$$D^n$ and $B^n$$B^n$ be closed $n$$n$-balls in $N$$N$ such that

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$D^n\subset{\rm Int}B^n$. Let $f:N\to\R^{2n-1}$$f:N\to\R^{2n-1}$ a general position immersion which is an embedding on the closure of $N-D^n$$N-D^n$. Let $N_0 := N-B^n$$N_0 := N-B^n$ and $f_0:=f|_{N-B^n}$$f_0:=f|_{N-B^n}$. We omit $\Z_{(n)}$$\Z_{(n)}$-coefficients. Then $f_0^{-1}f(D^n)$$f_0^{-1}f(D^n)$ represents a 1-cycle in $N$$N$. The homology class of this 1-cycle equals minus the normal Stiefel-Whitney class $PD\overline{W}^{n-1}(N)\in H_1(N)$$PD\overline{W}^{n-1}(N)\in H_1(N)$. This follows because

$\displaystyle 0 \overset{(1)}= [f^{-1}f'(N)] \overset{(2)}= [f_0^{-1}f'(N_0)]+[f_0^{-1}f'(B^n)] \overset{(3)} = PD\overline{W}^{n-1}(N)+[f_0^{-1}f(D^n)].$ $$\displaystyle 0 \overset{(1)}= [f^{-1}f'(N)] \overset{(2)}= [f_0^{-1}f'(N_0)]+[f_0^{-1}f'(B^n)] \overset{(3)} = PD\overline{W}^{n-1}(N)+[f_0^{-1}f(D^n)].$$

Here

• $f':N\to\R^{2n-1}$$f':N\to\R^{2n-1}$ is the map formed by a general position small length vector field normal to $f$$f$, and non-zero over $f(B^n)$$f(B^n)$;

• equality (1) holds because $f'$$f'$ is homotopic to a map whose image is disjoint from $f$$f$;

• equality (2) holds because $f_0$$f_0$-preimages of both $f'(N_0)$$f'(N_0)$ and $f'B^n$$f'B^n$ are 1-cycles, and $f|_{B^n}$$f|_{B^n}$-preimage of $f'(N)$$f'(N)$ is a 1-cycle which is null-homologous;

• equality (3) holds because the Euler class of a bundle is dual to the homology class of zeroes of a general position section, and because $[f_0^{-1}f'(B^n)] = [f_0^{-1}f'(D^n)] \overset{(4)}= [f_0^{-1}f(D^n)]$$[f_0^{-1}f'(B^n)] = [f_0^{-1}f'(D^n)] \overset{(4)}= [f_0^{-1}f(D^n)]$;

• equality (4) holds because $f(D^n)$$f(D^n)$ is homologous relative to the boundary and outside $f(N_0)$$f(N_0)$ to the union of $f'(D^n)$$f'(D^n)$ and the the normal vectors at points of $f(\partial D^n)$$f(\partial D^n)$.

6 A generalization to highly-connected manifolds

In this section let $N$$N$ be a closed orientable homologically $k$$k$-connected $n$$n$-manifold, $k\ge0$$k\ge0$. Recall the unknotting theorem [Skopenkov2016c, Theorem 2.4] that all embeddings $N \to\Rr^m$$N \to\Rr^m$ are isotopic when $m\ge 2n-k+1$$m\ge 2n-k+1$ and $n\ge2k+2$$n\ge2k+2$. In this section we generalize Theorem 2.1 to a description of $E^{2n-k}(N)$$E^{2n-k}(N)$ and further to $E^m(N)$$E^m(N)$ for $m\ge2n-2k+1$$m\ge2n-2k+1$.

6.1 Examples

Some simple examples are the Hudson tori $\Hud_{m,n,p}(a), \Hud'_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$$\Hud_{m,n,p}(a), \Hud'_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$.

Example 6.1 (cf. [Skopenkov2010, Definition 1.4]). Assume that $N$$N$ is $k$$k$-connected and $n-k\ge3$$n-k\ge3$. Then for an embedding $f_0:N\to\Rr^{2n-k}$$f_0:N\to\Rr^{2n-k}$ and a class $a\in H_{k+1}(N;\Zz)$$a\in H_{k+1}(N;\Zz)$ one can construct an embedding $f_a:N\to\Rr^{2n-k}$$f_a:N\to\Rr^{2n-k}$ by linked embedded connected sum analogously to the case $k=0$$k=0$ presented in Example 4.1.

We have $W(f_a,f_0)=a$$W(f_a,f_0)=a$ for the Whitney invariant. Hence by Theorem 6.2 below this construction gives a free transitive action of $H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$$H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$ on $E^{2n-k}(N)$$E^{2n-k}(N)$, provided $n\ge k+3$$n\ge k+3$ or $n\ge2k+4$$n\ge2k+4$ in the PL or smooth categories, respectively.

The embedding $f_a$$f_a$ has an alternative construction using parametric connected sum [Skopenkov2014, Remark 18.a].

6.2 Classification

Theorem 6.2. Let $N$$N$ be a closed oriented homologically $k$$k$-connected $n$$n$-manifold, $k\ge0$$k\ge0$. Then the Whitney invariant

$\displaystyle W:E^{2n-k}(N)\to H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$ $$\displaystyle W:E^{2n-k}(N)\to H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$$

is a bijection, provided $n\ge2k+4$$n\ge2k+4$ in the smooth category or $n\ge k+3$$n\ge k+3$ in the PL category.

This was proved for $k$$k$-connected manifolds in the smooth category [Haefliger&Hirsch1963, Theorem 2.4], and in the PL category in [Weber1967], [Hudson1969, $\S$$\S$11], cf. [Boechat&Haefliger1970, Theorem 1.6], [Boechat1971, Theorem 4.2], [Vrabec1977, Theorems 1.1 and 1.2], [Adachi1993, $\S$$\S$7]. The proof actually used the homological $k$$k$-connectedness assumption (basically because the $k$$k$-connectedness was used to ensure high enough connectedness of the complement in $\Rr^m$$\Rr^m$ to the image of $N$$N$, by Alexander duality and simple connectedness of the complement, so homological $k$$k$-connectedness of $N$$N$ is sufficient).

For $k=0$$k=0$ Theorem 6.2 is covered by Theorem 2.1; for $k\ge1$$k\ge1$ it is not. For $k+3\le n\le2k+1$$k+3\le n\le2k+1$ the PL case of Theorem 6.2 gives nothing but the Zeeman Unknotting Spheres Theorem [Skopenkov2016c, Theorem 2.3]. For the case of knotted tori see [Skopenkov2016k, Theorem 3.1].

An inverse to the map $W$$W$ of Theorem 6.2 is given by Example 6.1.

Because of the existence of knotted spheres the analogues of Theorem 6.2 for $n=k+2$$n=k+2$ in the PL case, and for $n\le2k+3$$n\le2k+3$ in the smooth case are false. So for the smooth category, $n\le2k+3$$n\le2k+3$ and $N$$N$ closed connected, a classification of $E^{2n-k}(N)$$E^{2n-k}(N)$ is much harder: for 40 years the only known complete readily calculable classification results were for homology spheres $N$$N$. E.g. $E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)}$$E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)}$ for any $s\ge2$$s\ge2$ [Haefliger1966, Corollary 8.14], [Skopenkov2016s, Theorem 3.2]. The following result for $n=2k+3$$n=2k+3$ was obtained using the Boéchat-Haefliger formula for the smoothing obstruction [Boechat&Haefliger1970, Theorem 2.1], [Boechat1971, Theorem 5.1]. Using that formula one can define the higher-dimensional Kreck invariant [Skopenkov2008, $\S$$\S$4].

Theorem 6.3 [Skopenkov2008, Higher-dimensional Classification Theorem]. Let $N$$N$ be a closed orientable homologically $(2l-2)$$(2l-2)$-connected $(4l-1)$$(4l-1)$-manifold. Then the Whitney invariant

$\displaystyle W:E^{6l}_D(N)\to H_{2l-1}(N)$ $$\displaystyle W:E^{6l}_D(N)\to H_{2l-1}(N)$$

is surjective and for any $u\in H_{2l-1}(N)$$u\in H_{2l-1}(N)$ the Kreck invariant

$\displaystyle \eta_u:W^{-1}u\to\Zz_{d(u)}$ $$\displaystyle \eta_u:W^{-1}u\to\Zz_{d(u)}$$

is a 1-1 correspondence, where $d(u)$$d(u)$ is the divisibility of the projection of $u$$u$ to the free part of $H_1(N)$$H_1(N)$.

Recall that the divisibility of zero is zero and the divisibility of $x\in G-\{0\}$$x\in G-\{0\}$ is $\max\{d\in\Zz\ | \ \text{there is }x_1\in G: \ x=dx_1\}$$\max\{d\in\Zz\ | \ \text{there is }x_1\in G: \ x=dx_1\}$.

How does one describe $E^m(N)$$E^m(N)$ when $N$$N$ is not $(2n-m)$$(2n-m)$-connected? For general $N$$N$ see the sentence on $E^{2n-1}(N)$$E^{2n-1}(N)$ at the end of $\S$$\S$2. We can say more as the connectivity $k$$k$ of $N$$N$ increases. Some estimations of $E^{2n-k-1}(N)$$E^{2n-k-1}(N)$ for a closed $k$$k$-connected $n$$n$-manifold $N$$N$ are presented in [Skopenkov2010]. For $k>1$$k>1$ one can go even further:

Theorem 6.4 [Becker&Glover1971, Corollary 1.3]. Let $N$$N$ be a closed $k$$k$-connected $n$$n$-manifold embeddable into $\Rr^m$$\Rr^m$, $m\ge2n-2k+1$$m\ge2n-2k+1$ and $2m\ge 3n+4$$2m\ge 3n+4$. Then there is a 1-1 correspondence

$\displaystyle E^m(N)\to [N_0, V_{m,n+1}].$ $$\displaystyle E^m(N)\to [N_0, V_{m,n+1}].$$

The 1-1 correspondence is constructed using `the Haefliger-Wu invariant' involving the configuration space of distinct pairs [Skopenkov2006, $\S$$\S$5]. For $k=0$$k=0$ Theorem 6.4 is the same as General Position Theorem [Skopenkov2016c, Theorem 2.1] (because $V_{2n+1,n+1}$$V_{2n+1,n+1}$ is $(n-1)$$(n-1)$-connected). For $k=1$$k=1$ Theorem 6.4 is covered by Theorem 6.2; for $k\ge2$$k\ge2$ it is not. For application to knotted tori see [Skopenkov2016k, Theorem 5.1]. For generalization to arbitrary manifolds see survey [Skopenkov2006, $\S$$\S$5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Observe that in Theorem 6.4 $V_{m,n+1}$$V_{m,n+1}$ can be replaced by $V_{M,M+n-m+1}$$V_{M,M+n-m+1}$ for any $M>n$$M>n$.

7 An orientation on the self-intersection set

Let $f:N\to\Rr^m$$f:N\to\Rr^m$ be a smooth map from an oriented $n$$n$-manifold $N$$N$ where $m\ge n+2$$m\ge n+2$. We assume that the closure

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${\rm Cl}\Sigma(f)$ of the self-intersection set of $f$$f$ has codimension 2 singularities, i.e., there is

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$P\subset{\rm Cl}\Sigma(f)$ such that

• both $P$$P$ and

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  ${\rm Cl}\Sigma(f)$ are subpolyhedra of some triangulation of $N$$N$,
• we have $\dim P\le\dim\Sigma(f)-2$$\dim P\le\dim\Sigma(f)-2$ and
• $\Sigma(f)-P$$\Sigma(f)-P$ is an open manifold consisting of self-transverse double points of $f$$f$.

Definition 7.1 (A canonical orientation on $\Sigma(f)-P$$\Sigma(f)-P$). Take points $x,y\in N$$x,y\in N$ away from $P$$P$ and such that $fx=fy$$fx=fy$. Then a $(2n-m)$$(2n-m)$-base $\xi_x$$\xi_x$ tangent to $\Sigma(f)-P$$\Sigma(f)-P$ at $x$$x$ gives a $(2n-m)$$(2n-m)$-base $\xi_y:=df_y^{-1}df_x(\xi_x)$$\xi_y:=df_y^{-1}df_x(\xi_x)$ tangent to $\Sigma(f)-P$$\Sigma(f)-P$ at $y$$y$. Since $N$$N$ is oriented, we can take positive $(m-n)$$(m-n)$-bases $\eta_x$$\eta_x$ and $\eta_y$$\eta_y$ at $x$$x$ and $y$$y$ normal to $\xi_x$$\xi_x$ and to $\xi_y$$\xi_y$. If the base $(df_x(\xi_x),df_x(\eta_x),df_y(\eta_y))$$(df_x(\xi_x),df_x(\eta_x),df_y(\eta_y))$ of $\Rr^m$$\Rr^m$ is positive, then call the base $\xi_x$$\xi_x$ positive. This is well-defined because a change of the sign of $\xi_x$$\xi_x$ forces changes of the signs of $\xi_y,\eta_x$$\xi_y,\eta_x$ and $\eta_y$$\eta_y$.

Remark 7.2 (Properties of the orientation just defined on $\Sigma(f) - P$$\Sigma(f) - P$)..

1. A change of the orientation of $N$$N$ forces changes of the signs of $\eta_x$$\eta_x$ and $\eta_y$$\eta_y$ and so does not change the orientation of $\Sigma(f)-P$$\Sigma(f)-P$.
2. The orientation on $\Sigma(f)-P$$\Sigma(f)-P$ need not extend to

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   ${\rm Cl}\Sigma(f)$: take the smooth cone $f:D^3\to\Rr^5$$f:D^3\to\Rr^5$ over a general position map $g:S^2\to\Rr^4$$g:S^2\to\Rr^4$ having only two transverse self-intersection points, where the smooth cone is defined by $f(tx):=(g(x)\sin(\pi t/2),\cos(\pi t/2))$$f(tx):=(g(x)\sin(\pi t/2),\cos(\pi t/2))$, for $x\in S^2$$x\in S^2$ and $t\in[0,1]$$t\in[0,1]$.
3. The orientation on $\Sigma(f)-P$$\Sigma(f)-P$ extends to

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   ${\rm Cl}\Sigma(f)$ if $m-n$$m-n$ is odd [Hudson1969, Lemma 11.4].

Remark 7.3 (A canonical orientation on $f\Sigma(f)-fP$$f\Sigma(f)-fP$ for $m-n$$m-n$ even). This remark is added as a complement for Definition 7.1 but is not needed for the definition of the Whitney invariant.

Take a $(2n-m)$$(2n-m)$-base $\xi$$\xi$ at a point $x\in f\Sigma(f)-fP$$x\in f\Sigma(f)-fP$. Since $N$$N$ is oriented, we can take a positive $(m-n)$$(m-n)$-base $\eta_+$$\eta_+$ normal to $f\Sigma(f)$$f\Sigma(f)$ in one sheet of $fN$$fN$. Analogously construct an $(m-n)$$(m-n)$-base $\eta_-$$\eta_-$ for the other sheet of $fN$$fN$. Since $m-n$$m-n$ is even, the orientation of the base $(\xi,\eta_+,\eta_-)$$(\xi,\eta_+,\eta_-)$ of $\Rr^m$$\Rr^m$ does not depend on choosing the first and the other sheet of $fN$$fN$ at $x$$x$. If the base $(\xi,\eta_+,\eta_-)$$(\xi,\eta_+,\eta_-)$ is positive, then call the base $\xi$$\xi$ positive. This is well-defined because a change of the sign of $\xi$$\xi$ forces changes of the signs of $\eta_+,\eta_-$$\eta_+,\eta_-$ and so of $(\xi,\eta_+,\eta_-)$$(\xi,\eta_+,\eta_-)$.

We remark that a change of the orientation of $N$$N$ forces changes of the signs of $\eta_+,\eta_-$$\eta_+,\eta_-$ and so does not change the orientation of $f\Sigma(f)-fP$$f\Sigma(f)-fP$.

8 References

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arxiv:math/0506464 MR2261638 (2007g:57049) Zbl 1113.57013

• [Tonkonog2010] D. Tonkonog, Embedding punctured $n$-manifolds in Euclidean $(2n-1)$-space
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• [Weber1967] C. Weber, Plongements de polyedres dans le domaine metastable, Comment. Math. Helv. 42 (1967), 1-27. MR0238330 (38 #6606) Zbl 0152.22402
• [Yasui1984] T. Yasui, Enumerating embeddings of $n$$n$-manifolds in Euclidean $(2n-1)$$(2n-1)$-space, J. Math. Soc. Japan 36 (1984), no.4, 555–576. MR759414 Zbl 0557.57019

, $\S]{Skopenkov2016c}. Denote N is a connected manifold of dimension $n>1$$n>1$, and $m \ge2n+1$$m \ge2n+1$, then every two embeddings $N \to\Rr^m$$N \to\Rr^m$ are isotopic [Skopenkov2016c, Theorem 3.2], [Skopenkov2006, Theorem 2.5]. In this page we summarize the situation for $m=2n\ge6$$m=2n\ge6$ and $N$$N$ is a connected, as well as in some more general situations. For the classification of embeddings of some disconnected manifolds see [Skopenkov2016h].

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$$\S$1, $\S$$\S$3]. Denote $1_n:=(1,0,\ldots,0)\in S^n$$1_n:=(1,0,\ldots,0)\in S^n$.

2 Classification

For the next theorem, the Whitney invariant $W$$W$ is defined in $\S$$\S$5 below.

Theorem 2.1. Assume that $N$$N$ is a closed connected $n$$n$-manifold, and either $n\ge4$$n\ge4$ or $n=3$$n=3$ and we are in the PL category.

(a) If $N$$N$ is oriented, the Whitney invariant,

$\displaystyle W:E^{2n}(N)\to H_1(N;\Zz_{\varepsilon(n-1)}),$ $$\displaystyle W:E^{2n}(N)\to H_1(N;\Zz_{\varepsilon(n-1)}),$$

is a 1-1 correspondence.

(b) If $N$$N$ is non-orientable, then there is a 1-1 correspondence

$\displaystyle E^{2n}(N)\to \begin{cases} H_1(N;\Zz_2) & n\text{ is odd}\\ \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s & n\text{ is even}\end{cases}.$ $$\displaystyle E^{2n}(N)\to \begin{cases} H_1(N;\Zz_2) & n\text{ is odd}\\ \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s & n\text{ is even}\end{cases}.$$

Remark 2.2 (Comments on the proof). Part (a) is proved in [Haefliger&Hirsch1963, Theorem 2.4] in the smooth category, and in [Weber1967, Theorem 4' in $\S$$\S$2], [Hudson1969, $\S$$\S$11], [Vrabec1977, Theorems 1.1 and 1.2] in the PL category, see also [Haefliger1962b, 1.3.e], [Haefliger1963], [Bausum1975, Theorem 43].

Part (b) is proved in [Bausum1975, Theorem 43] in the smooth category. By [Weber1967, Theorems 1 and 1' in $\S$$\S$2], [Skopenkov1997, Theorem 1.1.c] the proof works also in the PL category.

In Part (b) we replaced the kernel $\ker Sq^1$$\ker Sq^1$ from [Bausum1975, Theorem 43] by $\Zz_2^{s-1}$$\Zz_2^{s-1}$. This is possible because, as a specialist could see, $Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$$Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$ is given by multiplication with the first Stefel-Whitney class $w_1(N)$$w_1(N)$ (which equals to the first Wu class $v_1(N)$$v_1(N)$ [Milnor&Stasheff1974, Theorem 11.4]). Since $N$$N$ is non-orientable, $w_1(N)\neq 0$$w_1(N)\neq 0$. So by Poincaré duality, $\ker Sq^1 \cong \Zz_2^{s-1}$$\ker Sq^1 \cong \Zz_2^{s-1}$.

The 1-1 correspondence from (b) can presumably be defined as a (generalized) Whitney invariant, see [Vrabec1977], but the proof used 'the Haefliger-Wu invariant' whose definition can be found e.g. in [Skopenkov2006, $\S$$\S$5]. It would be interesting to check if part (b) is equivalent to different forms of description of $E^{2n}(N)$$E^{2n}(N)$ [Haefliger1962b, 1.3.e], [Haefliger1963], [Vrabec1977, Theorem 1.1].

The classification of smooth embeddings of 3-manifolds in $\Rr^6$$\Rr^6$ is more complicated, see Theorem 6.3 below for $l=1$$l=1$ or [Skopenkov2016t].

Concerning embeddings of connected $n$$n$-manifolds in $\Rr^{2n-1}$$\Rr^{2n-1}$ see [Yasui1984] for $n\ge5$$n\ge5$, [Skopenkov2016f] for $n=4$$n=4$, and [Saeki1999], [Skopenkov2010], [Tonkonog2010] for manifolds with boundary.

Theorem 2.1 is generalized to a description of $E^{2n-k}(N)$$E^{2n-k}(N)$ for closed $k$$k$-connected $n$$n$-manifolds $N$$N$, see Theorem 6.2.

3 Hudson and Lagrangian tori

Together with the Haefliger knotted sphere [Skopenkov2016t, Example 2.1], [Skopenkov2006, Example 3.4], the examples of Hudson tori presented below were the first examples of non-isotopic embeddings in codimension greater than 2. (Hudson's construction [Hudson1963] was not as explicit as those below.) Abbreviate

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${\rm i}_{2n,n-1}$ to just

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${\rm i}$.

Example 3.1. Let us construct, for any $a\in\Zz$$a\in\Zz$ and $n\ge2$$n\ge2$, a smooth embedding

$\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.$ $$\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.$$

We start with the cases $a=0,1$$a=0,1$.

Take the standard inclusion $\partial D^2\subset\partial D^{n+1}$$\partial D^2\subset\partial D^{n+1}$. The 'standard embedding' $\Hud_n(0)$$\Hud_n(0)$ is given by the standard inclusions

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$$\displaystyle \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\overset{{\rm i}}\to\Rr^{2n}.$$

Define the `standard embedding'

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$2{\rm i}:2D^{n+1}\times S^{n-1}\to\Rr^{2n}$ analogously to

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${\rm i}$, where $2$$2$ means homothety with coefficient 2.

Take the embedding $g_1$$g_1$ given by

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$$\displaystyle S^n=2\partial D^{n+1}\times1_{n-1}\subset 2D^{n+1}\times S^{n-1}\overset{2{\rm i}}\to\Rr^{2n}.$$

The segment

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$2{\rm i}([1,2]1_n\times1_{n-1})$ joins the images of $\Hud_n(0)$$\Hud_n(0)$ and $g_1$$g_1$; the interior of this segment misses the images. Let $\Hud_n(1)$$\Hud_n(1)$ be the linked embedded connected sum of $\Hud_n(0)$$\Hud_n(0)$ and $g_1$$g_1$ along this segment, compatible with the orientation, cf. [Avvakumov2017, $\S$$\S$1.5]. (Here 'linked' means that the images of the embeddings are not contained in disjoint cubes, unlike for the unlinked embedded connected sum [Skopenkov2016c, $\S$$\S$5].)

For $a\in\Zz$$a\in\Zz$ we repeat the above construction of $g_1$$g_1$ replacing $2\partial D^{n+1}\times1_{n-1}$$2\partial D^{n+1}\times1_{n-1}$ by $|a|$$|a|$ copies $(1+\frac1k)\partial D^{n+1}\times1_{n-1}$$(1+\frac1k)\partial D^{n+1}\times1_{n-1}$ of $S^n$$S^n$, $k=1,\ldots,|a|$$k=1,\ldots,|a|$. The copies are outside $D^{n+1}\times S^{n-1}$$D^{n+1}\times S^{n-1}$ and are `parallel' to $\partial D^{n+1}\times1_{n-1}$$\partial D^{n+1}\times1_{n-1}$. The copies have the standard orientation for $a>0$$a>0$ or the opposite orientation for $a<0$$a<0$. Then we make embedded connected sum along natural segments joining every $k$$k$-th copy to the $(k+1)$$(k+1)$-th copy. We obtain an embedding $g_a:S^n\to\Rr^{2n}$$g_a:S^n\to\Rr^{2n}$ which has disjoint images with $\Hud_n(0)$$\Hud_n(0)$. Let $\Hud_n(a)$$\Hud_n(a)$ be the linked embedded connected sum of $\Hud_n(0)$$\Hud_n(0)$ and $g_a$$g_a$.

The original motivation for Hudson was that $\Hud_n(1)$$\Hud_n(1)$ is not isotopic to $\Hud_n(0)$$\Hud_n(0)$ for any $n\ge3$$n\ge3$ (this is a particular case of Proposition 3.2 below). One might guess that $\Hud_n(a)$$\Hud_n(a)$ is not isotopic to $\Hud_n(a')$$\Hud_n(a')$ for $a\ne a'$$a\ne a'$ and that a $\Zz$$\Zz$-valued invariant of $E^{2n}(S^1 \times S^{n-1})$$E^{2n}(S^1 \times S^{n-1})$ can be defined by the homotopy class of the map

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$$\displaystyle S^n\overset g\to S^{2n}-{\rm i}(D^{n+1}\times S^{n-1})\sim S^{2n}-S^{n-1}\sim S^n \quad\text{which is}\quad a\in\pi_n(S^n)\cong\Zz.$$

However, this is only true for $n$$n$ odd.

Proposition 3.2. For $n\ge3$$n\ge3$ odd $\Hud_n(a)$$\Hud_n(a)$ is isotopic to $\Hud_n(a')$$\Hud_n(a')$ if and only if $a=a'$$a=a'$.

For $n\ge4$$n\ge4$ even $\Hud_n(a)$$\Hud_n(a)$ is isotopic to $\Hud_n(a')$$\Hud_n(a')$ if and only if $a\equiv a'\text{ mod}2$$a\equiv a'\text{ mod}2$.

Proposition 3.2 follows by calculation of the Whitney invariant (Remark 5.3.d below) and, for $n$$n$ even, by Theorem 2.1. This proposition holds with the same proof in the piecewise smooth category, whose definition is recalled in [Skopenkov2016f, Remark 1.1]). Proposition 3.2 also holds in the PL category (with an analogous construction of $\Hud_n(a)$$\Hud_n(a)$ for the PL category). It would be interesting to find an explicit construction of an isotopy between $\Hud_{2k}(a)$$\Hud_{2k}(a)$ and $\Hud_{2k}(a+2)$$\Hud_{2k}(a+2)$, cf. [Vrabec1977, $\S$$\S$5]. Analogously, $\Hud_2(a)$$\Hud_2(a)$ is not isotopic to $\Hud_2(a')$$\Hud_2(a')$ if $a\not\equiv a'\text{ mod}2$$a\not\equiv a'\text{ mod}2$. It would be interesting to know if the converse holds, e.g. is $\Hud_2(0)$$\Hud_2(0)$ (PS or smoothly) isotopic to $\Hud_2(2)$$\Hud_2(2)$.

Example 3.3. Take any $a\in\Zz$$a\in\Zz$. Take a map $\overline a:S^{n-1}\to S^{n-1}$$\overline a:S^{n-1}\to S^{n-1}$ of degree $a$$a$ (so we can take $\overline 1=\id$$\overline 1=\id$). Recall that $D^{n+1}=\{(y,x)\in D^n\times D^1\ :\ |y|^2+|x|^2\le1\}$$D^{n+1}=\{(y,x)\in D^n\times D^1\ :\ |y|^2+|x|^2\le1\}$. Define the smooth embedding $\Hud_n'(a)$$\Hud_n'(a)$ to be the composition

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$$\displaystyle S^1\times S^{n-1}\overset{\widehat a}\to D^{n+1}\times S^{n-1}\overset{\rm i}\to\Rr^{2n} \quad\text{where}\quad \widehat a(e^{i\theta},y):=(\overline a(y)\cos\theta,\sin\theta,y).$$

Figure 1: The Hudson torus

Let us present a geometric description of this embedding. Define a map $\widetilde a:S^0\times S^{n-1}\to S^{n-1}$$\widetilde a:S^0\times S^{n-1}\to S^{n-1}$ by $\widetilde a(s,t):=s\overline a(t)$$\widetilde a(s,t):=s\overline a(t)$. This map gives an embedding

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$$\displaystyle S^0\times S^{n-1}\xrightarrow{\widetilde a\times {\rm pr}_2}S^{n-1}\times S^{n-1}\subset D^{n+1}\times S^{n-1}\overset{{\rm i}}\to\Rr^{2n}.$$

See Figure 1. The image of

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$\widetilde a\times{\rm pr}_2$ is the union of the graphs of the maps $\overline a$$\overline a$ and $-\overline a$$-\overline a$. For any $t\in S^{n-1}$$t\in S^{n-1}$ the disk

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${\rm i}(D^{n+1}\times t)$ intersects the image at two points lying in

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${\rm i}(D^n\times t)$, i.e., at the image of an embedding

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$S^0\times t\to {\rm i}(D^n\times t)$. The embedding $\Hud_n'(a)$$\Hud_n'(a)$ is obtained by extending the latter embeddings to embeddings

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$S^1\times t\to {\rm i}(D^{n+1}\times t)$ for all $t$$t$. See Figure 2.

Figure 2: To the construction of the Hudson torus

Remark 3.4. (a) The analogue of Proposition 3.2 for $\Hud_n$$\Hud_n$ replaced to $\Hud_n'$$\Hud_n'$ holds, with an analogous proof.

(b) The embeddings $\Hud_n(a)$$\Hud_n(a)$ and $\Hud_n'(a)$$\Hud_n'(a)$ are smoothly isotopic for $n\ge4$$n\ge4$ and are PS isotopic for $n\ge3$$n\ge3$ [Skopenkov2006a, commutativity of the left upper square in the Restriction Lemma 5.2], [Skopenkov2015a, Lemma 2.15.c] (see [Skopenkov2016f, Remark 1.2]). This follows by calculation of the Whitney invariant (Remark 5.3.d below). It would be interesting to know if they are smoothly isotopic for $n=3$$n=3$. It would be interesting to know if they are piecewise smoothly isotopic for $n=2$$n=2$.

(b') Although construction of $\Hud_n(a)$$\Hud_n(a)$ and $\Hud_n'(a)$$\Hud_n'(a)$ is explicit, no explicit construction of an isotopy between them is known. Proof from (b) of their being isotopic is by calculating invariants, and using completeness result for the invariants. This proof is fairly constructive, so in principle it does give a specific isotopy. However, the construction is so complicated that one would not call such an isotopy explicit. This is analogous to the Smale `turning sphere inside out' theorem (although the Smale theorem concerns regular homotopy of immersions not isotopy of embeddings).

In higher dimensions, a proof that any embedding is isotopic to some of the constructed embeddings is even less likely to use an explicit construction. The only proofs we have go through calculating the invariants and using classification results.

(c) For $n=2$$n=2$ Example 3.3 gives what we call the 'left' Hudson torus. The 'right' Hudson torus is constructed analogously. It is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of $S^1\times S^1$$S^1\times S^1$. The right and the left Hudson tori are not isotopic by Remark 5.3.d below.

(d) Analogously one constructs the Hudson torus $\Hud_{n,p}(a):S^p\times S^{n-p}\to\Rr^{2n-p+1}$$\Hud_{n,p}(a):S^p\times S^{n-p}\to\Rr^{2n-p+1}$ for $a\in\Zz$$a\in\Zz$ and $n>p\ge0$$n>p\ge0$ or, more generally, $\Hud_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$$\Hud_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$ for $a\in\pi_n(S^{m-n+p-1})$$a\in\pi_n(S^{m-n+p-1})$ and $m>n>p\ge0$$m>n>p\ge0$. There are versions $\Hud'_{m,n,p}(a)$$\Hud'_{m,n,p}(a)$ of these constructions corresponding to Definition 3.3. For $p=0$$p=0$ this corresponds to the Zeeman map [Skopenkov2016h, Definition 2.2] and its composition with 'the unframed second Kirby move' [Skopenkov2015a, $\S$$\S$2.3]. It would be interesting to know if the links $\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$$\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$ are isotopic, cf. [Skopenkov2015a, Remark 2.7.b]. These constructions can be further generalized [Skopenkov2016k].

Example 3.5. Define a smooth embedding

$\displaystyle l:S^1\times S^{n-1}\to \C^n=\R^{2n}\quad\text{by}\quad l(z,x) := \left(1+\frac z2\right)x.$ $$\displaystyle l:S^1\times S^{n-1}\to \C^n=\R^{2n}\quad\text{by}\quad l(z,x) := \left(1+\frac z2\right)x.$$

Here $S^1\subset\R^2=\C$$S^1\subset\R^2=\C$ and $S^{n-1}\subset\R^n\subset\C^n$$S^{n-1}\subset\R^n\subset\C^n$. This embedding is known to be Lagrangian; see more in [Nemirovski2024].

4 Action by linked embedded connected sum

In this section we generalize the construction of the Hudson torus $\Hud(a)$$\Hud(a)$. Let $N$$N$ be a closed connected oriented $n$$n$-manifold. We work in the smooth category which we omit. Apparently analogous results hold for $n\ge3$$n\ge3$ in the PL and PS categories (see [Skopenkov2016f, Remark 1.2]).

Example 4.1. For any $n\ge4$$n\ge4$, an embedding $f_0:N\to\Rr^{2n}$$f_0:N\to\Rr^{2n}$ and $a\in H_1(N;\Zz)$$a\in H_1(N;\Zz)$, we shall construct an embedding $f_a:N\to\Rr^{2n}$$f_a:N\to\Rr^{2n}$. This embedding is said to be obtained by linked embedded connected sum of $f_0$$f_0$ with an $n$$n$-sphere representing the `homology Alexander dual' $A:=\widehat{A_{f_0}}a\in H_n(C_{f_0})$$A:=\widehat{A_{f_0}}a\in H_n(C_{f_0})$ of $a$$a$ (defined in [Skopenkov2005, Alexander Duality Lemma 4.6]).

Represent $a$$a$ by an embedding $a:S^1\to N$$a:S^1\to N$. By definition, the class $A$$A$ is represented by properly oriented $\nu_{f_0}^{-1}a(S^1)$$\nu_{f_0}^{-1}a(S^1)$. Since any orientable bundle over $S^1$$S^1$ is trivial, $\nu_{f_0}^{-1}a(S^1)\cong S^1\times S^{n-1}$$\nu_{f_0}^{-1}a(S^1)\cong S^1\times S^{n-1}$. Take an embedding $g:S^1\times S^{n-1}\to C_{f_0}$$g:S^1\times S^{n-1}\to C_{f_0}$ whose image is $\nu_{f_0}^{-1}a(S^1)$$\nu_{f_0}^{-1}a(S^1)$ and which represents $A$$A$. By embedded surgery on $S^1\times1_{n-1}\subset S^1\times S^{n-1}$$S^1\times1_{n-1}\subset S^1\times S^{n-1}$ we obtain an embedding $g_1:S^n\to C_{f_0}$$g_1:S^n\to C_{f_0}$ representing $A$$A$ (see details in Proposition 4.2 below). Define $f_a$$f_a$ to be the linked embedded connected sum of $f_0$$f_0$ and $g_1$$g_1$, along some arc joining their images.

Proposition 4.2 (Embedded surgery). For any $n\ge3$$n\ge3$, a neighborhood $U$$U$ of a codimension at least 3 subpolyhedron in $\Rr^{2n}$$\Rr^{2n}$ and an embedding $g:S^1\times S^{n-1}\to\Rr^{2n}-U$$g:S^1\times S^{n-1}\to\Rr^{2n}-U$ there is an embedding $g_1:S^n\to\Rr^{2n}-U$$g_1:S^n\to\Rr^{2n}-U$ homologous to $g$$g$.

Proof. Take a vector field on $g(S^1\times1_{n-1})$$g(S^1\times1_{n-1})$ normal to $g(S^1\times S^{n-1})$$g(S^1\times S^{n-1})$. Extend $g|_{S^1\times1_{n-1}}$$g|_{S^1\times1_{n-1}}$ along this vector field to a map $\overline b:D^2\to\Rr^{2n}$$\overline b:D^2\to\Rr^{2n}$.

Since $2n>4$$2n>4$ and $U$$U$ is a neighborhood $U$$U$ of a codimension at least 3 subpolyhedron, by general position we may assume that $\overline b$$\overline b$ is an embedding and that

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$\overline b({\rm Int}D^2)$ misses $U\cup g(S^1\times S^{n-1})$$U\cup g(S^1\times S^{n-1})$.

Since $n-1>1$$n-1>1$, we have $\pi_1(V_{2n-2,n-1})=0$$\pi_1(V_{2n-2,n-1})=0$. Hence the standard $(n-1)$$(n-1)$-framing of $S^1\times1_{n-1}$$S^1\times1_{n-1}$ in $S^1\times S^{n-1}$$S^1\times S^{n-1}$ extends to an $(n-1)$$(n-1)$-framing on $\overline b(D^2)$$\overline b(D^2)$ in $\Rr^{2n}$$\Rr^{2n}$. Thus $\overline b$$\overline b$ extends to an embedding

$\displaystyle \widehat b:D^2\times D^{n-1}\to\R^{2n}-U\quad\text{such that}\quad \widehat b(\partial D^2\times D^{n-1})\subset g(S^1\times S^{n-1}).$ $$\displaystyle \widehat b:D^2\times D^{n-1}\to\R^{2n}-U\quad\text{such that}\quad \widehat b(\partial D^2\times D^{n-1})\subset g(S^1\times S^{n-1}).$$

Take an embedding $g_1:S^n\to\R^{2n}-U$$g_1:S^n\to\R^{2n}-U$ such that

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$$\displaystyle g_1(S^n)\ =\ g(S^1\times S^{n-1})-\widehat b(\partial D^2\times{\rm Int}D^{n-1}) \bigcup\limits_{\widehat b(\partial D^2\times\partial D^{n-1})} \widehat b(D^2\times\partial D^{n-1})\ \cong\ S^n.$$

with proper orientation so that $g_1$$g_1$ is homologous to $g$$g$. QED

The isotopy class of the embedding $f_a$$f_a$ is independent of the choises in the construction. The independence of the arc and of the maps $g,g_1$$g,g_1$ follows by $n\ge3$$n\ge3$ and by Proposition 4.3 below, respectively.

By Definition 5.1 of the Whitney invariant, $W(f_a,f_0)$$W(f_a,f_0)$ is $a$$a$ for $n\ge3$$n\ge3$ odd and $a \mod2$$a \mod2$ for $n\ge4$$n\ge4$ even. Thus by Theorem 2.1.a for $n\ge4$$n\ge4$ all isotopy classes of embeddings $N\to\Rr^{2n}$$N\to\Rr^{2n}$ can be obtained from any chosen embedding $f_0$$f_0$ by the above construction.

Proposition 4.3. For any $n\ge4$$n\ge4$ both the linked embedded connected sum and parametric connected sum (introduced in [Skopenkov2006a], [Skopenkov2015a]) define free transitive actions of $H_1(N;\Zz_{\varepsilon(n-1)})$$H_1(N;\Zz_{\varepsilon(n-1)})$ on $E^{2n}(N)$$E^{2n}(N)$.

This follows by Theorem 2.1.a and by [Skopenkov2014, Remark 18.a].

5 The Whitney invariant

Let $N$$N$ be a closed $n$$n$-manifold. Take an embedding $f_0:N\to\Rr^m$$f_0:N\to\Rr^m$. Fix an orientation on $\Rr^m$$\Rr^m$. For any other embedding $f \colon N \to \Rr^m$$f \colon N \to \Rr^m$ we define the Whitney invariant

$\displaystyle W(f, f_0)=W_{f_0}(f)=W(f)\in H_{2n-m+1}(N;\Zz_N).$ $$\displaystyle W(f, f_0)=W_{f_0}(f)=W(f)\in H_{2n-m+1}(N;\Zz_N).$$

Here the coefficients $\Zz_N$$\Zz_N$ are $\Zz$$\Zz$ if $N$$N$ is oriented and $m-n$$m-n$ is odd, and are $\Zz_2$$\Zz_2$ otherwise.

Roughly speaking,

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$W(f):=[{\rm Cl}\Sigma(H)]$ is defined as the homology class of the closure of the self-intersection set of a general position homotopy $H$$H$ between $f$$f$ and $f_0$$f_0$.

This is formalized in Definition 5.2 in the smooth category, following [Skopenkov2010], see also [Haefliger&Hirsch1963]. The definition in the PL category is analogous [Hudson1969, $\S$$\S$11], [Vrabec1977, p. 145], [Skopenkov2006, $\S$$\S$2.4 `The Whitney invariant']. We begin by presenting a simpler definition, Definition 5.1, for a particular case.

For Theorem 2.1 only the case $m=2n$$m=2n$ is required.

Definition 5.1. Assume that $N$$N$ is $(2n-m)$$(2n-m)$-connected and $2m\ge3n+3$$2m\ge3n+3$. Then by [Haefliger&Hirsch1963, Theorem 3.1.b] restrictions of $f$$f$ and $f_0$$f_0$ to $N_0$$N_0$ are isotopic, cf. [Takase2006, Lemma 2.2]. (Here is sketch of an argument. Using the Smale-Hirsch classification of immersions we obtain that restrictions of $f$$f$ and $f_0$$f_0$ to $N_0$$N_0$ are `regular homotopic', see [Koschorke2013, Definition 2.7]. Since $N$$N$ is $(2n-m)$$(2n-m)$-connected, $N_0$$N_0$ retracts to an $(m-n-1)$$(m-n-1)$-dimensional polyhedron. Therefore these restrictions are isotopic.)

So we can make an isotopy of $f$$f$ and assume that $f=f_0$$f=f_0$ on $N_0$$N_0$. Take a general position homotopy $F:B^n\times I\to\Rr^m$$F:B^n\times I\to\Rr^m$ relative to $\partial B^n$$\partial B^n$ between the restrictions of $f$$f$ and $f_0$$f_0$ to $B^n$$B^n$. Let $f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$$f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$ (`the intersection of this homotopy with $f(N-B^n)$$f(N-B^n)$').

Since $n+2(n+1)<2m$$n+2(n+1)<2m$, by general position

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${\rm Cl}(f\cap F)$ is a compact $(2n+1-m)$$(2n+1-m)$-manifold whose boundary is contained in $\partial N_0$$\partial N_0$.

So $f\cap F$$f\cap F$ carries a homology class with $\Zz_2$$\Zz_2$ coefficients. If $m-n$$m-n$ is odd and $N$$N$ is oriented, then $f\cap F$$f\cap F$ has a natural orientation defined below, and so carries a homology class with $\Zz$$\Zz$ coefficients. Define $W(f)$$W(f)$ to be the homology class:

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$$\displaystyle W(f):=[{\rm Cl}(f\cap F)]\in H_{2n-m+1}(N_0,\partial N_0;\Zz_N)\cong H_{2n-m+1}(N;\Zz_N).$$

The orientation on $f\cap F$$f\cap F$ (extendable to

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${\rm Cl}(f\cap F)$) is defined (for $m-n$$m-n$ odd and $N$$N$ is oriented) as follows (cf. Remark 7.3). For any point $x_f\in f\cap F$$x_f\in f\cap F$ take a base $\xi_f$$\xi_f$ at $x_f$$x_f$ tangent to $f\cap F$$f\cap F$. Complete this base to a positive base $(\xi_f,\eta_f)$$(\xi_f,\eta_f)$ tangent to $N$$N$. Since $n+2(n+1)<2m$$n+2(n+1)<2m$, by general position there is a unique point $x_F\in B^n\times I$$x_F\in B^n\times I$ such that $Fx_F=fx_f$$Fx_F=fx_f$. The tangent base $\xi_f$$\xi_f$ at $x_f$$x_f$ thus gives a base $\xi_F$$\xi_F$ at $x_F$$x_F$ tangent to $B^n\times I$$B^n\times I$ such that $df(x_f)\xi_f=dF(x_F)\xi_F$$df(x_f)\xi_f=dF(x_F)\xi_F$. Complete this base $\xi_F$$\xi_F$ to a positive base $(\xi_F,\eta_F)$$(\xi_F,\eta_F)$ tangent to $B^n\times I$$B^n\times I$, where the orientation on $B^n$$B^n$ comes from $N$$N$. The union $\zeta=(df(x_f)\xi_f,df(x_f)\eta_f,dF(x_F)\eta_F)$$\zeta=(df(x_f)\xi_f,df(x_f)\eta_f,dF(x_F)\eta_F)$ of the images of the constructed bases is a base at $fx_f=Fx_F$$fx_f=Fx_F$ of $\Rr^m$$\Rr^m$. If $\zeta$$\zeta$ is positive, then call the tangent base $\xi_f$$\xi_f$ of $f\cap F$$f\cap F$ `positive'. Since a change of the orientation on $f\cap F$$f\cap F$ forces a change of the orientation of $\zeta$$\zeta$, this condition indeed defines an orientation on $f\cap F$$f\cap F$.

Definition 5.2. Assume that $m\ge n+2$$m\ge n+2$. Take a general position homotopy $H:N\times I\to\Rr^m\times I$$H:N\times I\to\Rr^m\times I$ between $f_0$$f_0$ and $f$$f$.

The closure

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${\rm Cl}\Sigma(H)$ of the self-intersection set carries a cycle mod 2. If $N$$N$ is oriented and $m-n$$m-n$ is odd, the closure also carries an integer cycle. See [Hudson1967, $\S$$\S$11], [Skopenkov2006, $\S$$\S$2.3 `The Whitney obstruction']. Let us informally explain these facts. For $2m\ge3n+2$$2m\ge3n+2$ by general position the closure

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${\rm Cl}\Sigma(H)$ can be assumed to be a submanifold. In general, since $m\ge n+2$$m\ge n+2$, by general position the closure has codimension 2 singularities, as defined in $\S$$\S$7. So the closure carries a cycle mod 2. When $m-n$$m-n$ is odd the closure also has a canonical orientation (see Definition 7.1 and Remark 7.2), so the closure carries an integer cycle.

Define the Whitney invariant to be the homology class:

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$$\displaystyle W(f):=[{\rm Cl}\Sigma(H)]\in H_{2n-m+1}(N\times I;\Zz_N)\cong H_{2n-m+1}(N;\Zz_N).$$

Clearly, $W(f) = W(f')$$W(f) = W(f')$ if $f$$f$ is isotopic to $f'$$f'$. Hence the Whitney invariant defines a map

$\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_N),\quad [f] \mapsto W(f).$ $$\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_N),\quad [f] \mapsto W(f).$$

Clearly, $W(f_0)=0$$W(f_0)=0$ (for both definitions).

The definition of $W$$W$ depends on the choice of $f_0$$f_0$, but we write $W$$W$ not $W_{f_0}$$W_{f_0}$ for brevity.

Remark 5.3. (a) The Whitney invariant is well-defined by Definition 5.2, i.e. is independent of the choice of a general position homotopy $H:N\times I\to\Rr^m\times I$$H:N\times I\to\Rr^m\times I$ from $f_0$$f_0$ to $f$$f$.

This follows from the equality

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$[{\rm Cl}\Sigma(H_0)]−[{\rm Cl}\Sigma(H_1)] = \partial [{\rm Cl}\Sigma(H_{01})]$ for a general position homotopy $H_{01}:N\times I\times I\to\Rr^m\times I\times I$$H_{01}:N\times I\times I\to\Rr^m\times I\times I$ between general position homotopies $H_0,H_1:N\times I\to\Rr^m\times I$$H_0,H_1:N\times I\to\Rr^m\times I$ from $f_0$$f_0$ to $f$$f$. See details in [Hudson1969, $\S$$\S$11].

(b) Definition 5.1 is a particular case of Definition 5.2. (Indeed, if $f=f_0$$f=f_0$ on $N_0$$N_0$, we can take $H$$H$ to be fixed on $N_0$$N_0$. See details in [Skopenkov2010, Difference Lemma 2.4].) Hence the Whitney invariant is well-defined by Definition 5.1, i.e. independent of the choice of $F$$F$ and of the isotopy making $f=f_0$$f=f_0$ outside $B^n$$B^n$.

(c) The class $W(f)$$W(f)$ is independent of the choice of the orientation on $N$$N$ (because a change of the orientation on $N$$N$ forces a change of the orientation on $f\cap F$$f\cap F$ or on

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${\rm Cl}\Sigma(H)$). For the reflection $\sigma:\Rr^m\to\Rr^m$$\sigma:\Rr^m\to\Rr^m$ with respect to a hyperplane we have $W(\sigma\circ f)=-W(f)$$W(\sigma\circ f)=-W(f)$ (because a change of the orientation on $\Rr^m$$\Rr^m$ forces a change of the orientation on $f\cap F$$f\cap F$ or on

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${\rm Cl}\Sigma(H)$; for Definition 5.1 also observe that we may assume that $f=f_0=\sigma\circ f$$f=f_0=\sigma\circ f$ on $N_0$$N_0$).

(d) For the Hudson tori $W(\Hud_n(a))=W(\Hud'_n(a))$$W(\Hud_n(a))=W(\Hud'_n(a))$ is $a$$a$ or $a\mod2$$a\mod2$ for $n\ge3$$n\ge3$, and $W(\Hud'_2(a))=(a\mod2,0)$$W(\Hud'_2(a))=(a\mod2,0)$.

For $\Hud_n(a)$$\Hud_n(a)$ this is clear by Definition 5.1. For $\Hud'_n(a)$$\Hud'_n(a)$ and $n\ge3$$n\ge3$ this was proved in [Hudson1963] (with a different but equivalent definition of the Whitney invariant; using and proving a particular case of Remark 5.3.f). For $\Hud'_2(a)$$\Hud'_2(a)$ the proof is analogous.

(e) $W(f\#g)=W(f)$$W(f\#g)=W(f)$ for any pair of embeddings $f:N\to\Rr^m$$f:N\to\Rr^m$ and $g:S^n\to\Rr^m$$g:S^n\to\Rr^m$. This is clear by Definition 5.1 because $W(f\#g)-W(f)=W(f\#g,f_0)-W(f,f_0)=W(f\#g,f)=0$$W(f\#g)-W(f)=W(f\#g,f_0)-W(f,f_0)=W(f\#g,f)=0$. Let us prove the latter equality. Take the identical isotopy $H_f$$H_f$ of $f$$f$ and a general position homotopy $H_g$$H_g$ between $g$$g$ and the standard embedding. Then the boundary connected sum $H_f\sharp H_g$$H_f\sharp H_g$ is a general position homotopy between $f\#g$$f\#g$ and an embedding isotopic to $f$$f$. The cycle

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${\rm Cl}\Sigma(H_f\sharp H_g)={\rm Cl}\Sigma H_g$ is null-homologous in $S^n$$S^n$ and hence in $N\cong N\#S^n$$N\cong N\#S^n$; cf. [Skopenkov2008, Addendum to the Classification Theorem].

(f) For $m=2n+1$$m=2n+1$ and $N=S^n\sqcup S^n$$N=S^n\sqcup S^n$ the Whitney invariant equals to the pair of linking coefficients [Skopenkov2016h, $\S$$\S$3].

(g) The Whitney invariant need not be a bijection for $m<2n$$m<2n$. This is seen, for example, by applying Theorem 6.4 below in case of knotted tori [Skopenkov2016k, Theorem 5.1]) or by taking $n$$n$ even, $N$$N$ non-orientable, $m=2n$$m=2n$ and applying by Theorem 2.1.b.

(h) The paper [Haefliger&Hirsch1963, $\S$$\S$2] gives yet another equivalent definition of the Whitney invariant, and proves its injectivity under certain restrictions (the Haefliger-Hirsch $h$$h$-principle, see Theorem 6.2).

(i) Let $N$$N$ be a closed orientable $n$$n$-manifold. For two embeddings $f,g:N\to\R^{2n-1}$$f,g:N\to\R^{2n-1}$ the invariant defined in [Haefliger&Hirsch1963, $\S$$\S$2] equals plus or minus their Whitney invariant $W(f,g)$$W(f,g)$ from Definition 5.1. This is proved analogously to the following version.

Let $D^n$$D^n$ and $B^n$$B^n$ be closed $n$$n$-balls in $N$$N$ such that

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$D^n\subset{\rm Int}B^n$. Let $f:N\to\R^{2n-1}$$f:N\to\R^{2n-1}$ a general position immersion which is an embedding on the closure of $N-D^n$$N-D^n$. Let $N_0 := N-B^n$$N_0 := N-B^n$ and $f_0:=f|_{N-B^n}$$f_0:=f|_{N-B^n}$. We omit $\Z_{(n)}$$\Z_{(n)}$-coefficients. Then $f_0^{-1}f(D^n)$$f_0^{-1}f(D^n)$ represents a 1-cycle in $N$$N$. The homology class of this 1-cycle equals minus the normal Stiefel-Whitney class $PD\overline{W}^{n-1}(N)\in H_1(N)$$PD\overline{W}^{n-1}(N)\in H_1(N)$. This follows because

$\displaystyle 0 \overset{(1)}= [f^{-1}f'(N)] \overset{(2)}= [f_0^{-1}f'(N_0)]+[f_0^{-1}f'(B^n)] \overset{(3)} = PD\overline{W}^{n-1}(N)+[f_0^{-1}f(D^n)].$ $$\displaystyle 0 \overset{(1)}= [f^{-1}f'(N)] \overset{(2)}= [f_0^{-1}f'(N_0)]+[f_0^{-1}f'(B^n)] \overset{(3)} = PD\overline{W}^{n-1}(N)+[f_0^{-1}f(D^n)].$$

Here

• $f':N\to\R^{2n-1}$$f':N\to\R^{2n-1}$ is the map formed by a general position small length vector field normal to $f$$f$, and non-zero over $f(B^n)$$f(B^n)$;

• equality (1) holds because $f'$$f'$ is homotopic to a map whose image is disjoint from $f$$f$;

• equality (2) holds because $f_0$$f_0$-preimages of both $f'(N_0)$$f'(N_0)$ and $f'B^n$$f'B^n$ are 1-cycles, and $f|_{B^n}$$f|_{B^n}$-preimage of $f'(N)$$f'(N)$ is a 1-cycle which is null-homologous;

• equality (3) holds because the Euler class of a bundle is dual to the homology class of zeroes of a general position section, and because $[f_0^{-1}f'(B^n)] = [f_0^{-1}f'(D^n)] \overset{(4)}= [f_0^{-1}f(D^n)]$$[f_0^{-1}f'(B^n)] = [f_0^{-1}f'(D^n)] \overset{(4)}= [f_0^{-1}f(D^n)]$;

• equality (4) holds because $f(D^n)$$f(D^n)$ is homologous relative to the boundary and outside $f(N_0)$$f(N_0)$ to the union of $f'(D^n)$$f'(D^n)$ and the the normal vectors at points of $f(\partial D^n)$$f(\partial D^n)$.

6 A generalization to highly-connected manifolds

In this section let $N$$N$ be a closed orientable homologically $k$$k$-connected $n$$n$-manifold, $k\ge0$$k\ge0$. Recall the unknotting theorem [Skopenkov2016c, Theorem 2.4] that all embeddings $N \to\Rr^m$$N \to\Rr^m$ are isotopic when $m\ge 2n-k+1$$m\ge 2n-k+1$ and $n\ge2k+2$$n\ge2k+2$. In this section we generalize Theorem 2.1 to a description of $E^{2n-k}(N)$$E^{2n-k}(N)$ and further to $E^m(N)$$E^m(N)$ for $m\ge2n-2k+1$$m\ge2n-2k+1$.

6.1 Examples

Some simple examples are the Hudson tori $\Hud_{m,n,p}(a), \Hud'_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$$\Hud_{m,n,p}(a), \Hud'_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$.

Example 6.1 (cf. [Skopenkov2010, Definition 1.4]). Assume that $N$$N$ is $k$$k$-connected and $n-k\ge3$$n-k\ge3$. Then for an embedding $f_0:N\to\Rr^{2n-k}$$f_0:N\to\Rr^{2n-k}$ and a class $a\in H_{k+1}(N;\Zz)$$a\in H_{k+1}(N;\Zz)$ one can construct an embedding $f_a:N\to\Rr^{2n-k}$$f_a:N\to\Rr^{2n-k}$ by linked embedded connected sum analogously to the case $k=0$$k=0$ presented in Example 4.1.

We have $W(f_a,f_0)=a$$W(f_a,f_0)=a$ for the Whitney invariant. Hence by Theorem 6.2 below this construction gives a free transitive action of $H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$$H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$ on $E^{2n-k}(N)$$E^{2n-k}(N)$, provided $n\ge k+3$$n\ge k+3$ or $n\ge2k+4$$n\ge2k+4$ in the PL or smooth categories, respectively.

The embedding $f_a$$f_a$ has an alternative construction using parametric connected sum [Skopenkov2014, Remark 18.a].

6.2 Classification

Theorem 6.2. Let $N$$N$ be a closed oriented homologically $k$$k$-connected $n$$n$-manifold, $k\ge0$$k\ge0$. Then the Whitney invariant

$\displaystyle W:E^{2n-k}(N)\to H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$ $$\displaystyle W:E^{2n-k}(N)\to H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$$

is a bijection, provided $n\ge2k+4$$n\ge2k+4$ in the smooth category or $n\ge k+3$$n\ge k+3$ in the PL category.

This was proved for $k$$k$-connected manifolds in the smooth category [Haefliger&Hirsch1963, Theorem 2.4], and in the PL category in [Weber1967], [Hudson1969, $\S$$\S$11], cf. [Boechat&Haefliger1970, Theorem 1.6], [Boechat1971, Theorem 4.2], [Vrabec1977, Theorems 1.1 and 1.2], [Adachi1993, $\S$$\S$7]. The proof actually used the homological $k$$k$-connectedness assumption (basically because the $k$$k$-connectedness was used to ensure high enough connectedness of the complement in $\Rr^m$$\Rr^m$ to the image of $N$$N$, by Alexander duality and simple connectedness of the complement, so homological $k$$k$-connectedness of $N$$N$ is sufficient).

For $k=0$$k=0$ Theorem 6.2 is covered by Theorem 2.1; for $k\ge1$$k\ge1$ it is not. For $k+3\le n\le2k+1$$k+3\le n\le2k+1$ the PL case of Theorem 6.2 gives nothing but the Zeeman Unknotting Spheres Theorem [Skopenkov2016c, Theorem 2.3]. For the case of knotted tori see [Skopenkov2016k, Theorem 3.1].

An inverse to the map $W$$W$ of Theorem 6.2 is given by Example 6.1.

Because of the existence of knotted spheres the analogues of Theorem 6.2 for $n=k+2$$n=k+2$ in the PL case, and for $n\le2k+3$$n\le2k+3$ in the smooth case are false. So for the smooth category, $n\le2k+3$$n\le2k+3$ and $N$$N$ closed connected, a classification of $E^{2n-k}(N)$$E^{2n-k}(N)$ is much harder: for 40 years the only known complete readily calculable classification results were for homology spheres $N$$N$. E.g. $E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)}$$E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)}$ for any $s\ge2$$s\ge2$ [Haefliger1966, Corollary 8.14], [Skopenkov2016s, Theorem 3.2]. The following result for $n=2k+3$$n=2k+3$ was obtained using the Boéchat-Haefliger formula for the smoothing obstruction [Boechat&Haefliger1970, Theorem 2.1], [Boechat1971, Theorem 5.1]. Using that formula one can define the higher-dimensional Kreck invariant [Skopenkov2008, $\S$$\S$4].

Theorem 6.3 [Skopenkov2008, Higher-dimensional Classification Theorem]. Let $N$$N$ be a closed orientable homologically $(2l-2)$$(2l-2)$-connected $(4l-1)$$(4l-1)$-manifold. Then the Whitney invariant

$\displaystyle W:E^{6l}_D(N)\to H_{2l-1}(N)$ $$\displaystyle W:E^{6l}_D(N)\to H_{2l-1}(N)$$

is surjective and for any $u\in H_{2l-1}(N)$$u\in H_{2l-1}(N)$ the Kreck invariant

$\displaystyle \eta_u:W^{-1}u\to\Zz_{d(u)}$ $$\displaystyle \eta_u:W^{-1}u\to\Zz_{d(u)}$$

is a 1-1 correspondence, where $d(u)$$d(u)$ is the divisibility of the projection of $u$$u$ to the free part of $H_1(N)$$H_1(N)$.

Recall that the divisibility of zero is zero and the divisibility of $x\in G-\{0\}$$x\in G-\{0\}$ is $\max\{d\in\Zz\ | \ \text{there is }x_1\in G: \ x=dx_1\}$$\max\{d\in\Zz\ | \ \text{there is }x_1\in G: \ x=dx_1\}$.

How does one describe $E^m(N)$$E^m(N)$ when $N$$N$ is not $(2n-m)$$(2n-m)$-connected? For general $N$$N$ see the sentence on $E^{2n-1}(N)$$E^{2n-1}(N)$ at the end of $\S$$\S$2. We can say more as the connectivity $k$$k$ of $N$$N$ increases. Some estimations of $E^{2n-k-1}(N)$$E^{2n-k-1}(N)$ for a closed $k$$k$-connected $n$$n$-manifold $N$$N$ are presented in [Skopenkov2010]. For $k>1$$k>1$ one can go even further:

Theorem 6.4 [Becker&Glover1971, Corollary 1.3]. Let $N$$N$ be a closed $k$$k$-connected $n$$n$-manifold embeddable into $\Rr^m$$\Rr^m$, $m\ge2n-2k+1$$m\ge2n-2k+1$ and $2m\ge 3n+4$$2m\ge 3n+4$. Then there is a 1-1 correspondence

$\displaystyle E^m(N)\to [N_0, V_{m,n+1}].$ $$\displaystyle E^m(N)\to [N_0, V_{m,n+1}].$$

The 1-1 correspondence is constructed using `the Haefliger-Wu invariant' involving the configuration space of distinct pairs [Skopenkov2006, $\S$$\S$5]. For $k=0$$k=0$ Theorem 6.4 is the same as General Position Theorem [Skopenkov2016c, Theorem 2.1] (because $V_{2n+1,n+1}$$V_{2n+1,n+1}$ is $(n-1)$$(n-1)$-connected). For $k=1$$k=1$ Theorem 6.4 is covered by Theorem 6.2; for $k\ge2$$k\ge2$ it is not. For application to knotted tori see [Skopenkov2016k, Theorem 5.1]. For generalization to arbitrary manifolds see survey [Skopenkov2006, $\S$$\S$5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Observe that in Theorem 6.4 $V_{m,n+1}$$V_{m,n+1}$ can be replaced by $V_{M,M+n-m+1}$$V_{M,M+n-m+1}$ for any $M>n$$M>n$.

7 An orientation on the self-intersection set

Let $f:N\to\Rr^m$$f:N\to\Rr^m$ be a smooth map from an oriented $n$$n$-manifold $N$$N$ where $m\ge n+2$$m\ge n+2$. We assume that the closure

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${\rm Cl}\Sigma(f)$ of the self-intersection set of $f$$f$ has codimension 2 singularities, i.e., there is

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$P\subset{\rm Cl}\Sigma(f)$ such that

• both $P$$P$ and

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  ${\rm Cl}\Sigma(f)$ are subpolyhedra of some triangulation of $N$$N$,
• we have $\dim P\le\dim\Sigma(f)-2$$\dim P\le\dim\Sigma(f)-2$ and
• $\Sigma(f)-P$$\Sigma(f)-P$ is an open manifold consisting of self-transverse double points of $f$$f$.

Definition 7.1 (A canonical orientation on $\Sigma(f)-P$$\Sigma(f)-P$). Take points $x,y\in N$$x,y\in N$ away from $P$$P$ and such that $fx=fy$$fx=fy$. Then a $(2n-m)$$(2n-m)$-base $\xi_x$$\xi_x$ tangent to $\Sigma(f)-P$$\Sigma(f)-P$ at $x$$x$ gives a $(2n-m)$$(2n-m)$-base $\xi_y:=df_y^{-1}df_x(\xi_x)$$\xi_y:=df_y^{-1}df_x(\xi_x)$ tangent to $\Sigma(f)-P$$\Sigma(f)-P$ at $y$$y$. Since $N$$N$ is oriented, we can take positive $(m-n)$$(m-n)$-bases $\eta_x$$\eta_x$ and $\eta_y$$\eta_y$ at $x$$x$ and $y$$y$ normal to $\xi_x$$\xi_x$ and to $\xi_y$$\xi_y$. If the base $(df_x(\xi_x),df_x(\eta_x),df_y(\eta_y))$$(df_x(\xi_x),df_x(\eta_x),df_y(\eta_y))$ of $\Rr^m$$\Rr^m$ is positive, then call the base $\xi_x$$\xi_x$ positive. This is well-defined because a change of the sign of $\xi_x$$\xi_x$ forces changes of the signs of $\xi_y,\eta_x$$\xi_y,\eta_x$ and $\eta_y$$\eta_y$.

Remark 7.2 (Properties of the orientation just defined on $\Sigma(f) - P$$\Sigma(f) - P$)..

1. A change of the orientation of $N$$N$ forces changes of the signs of $\eta_x$$\eta_x$ and $\eta_y$$\eta_y$ and so does not change the orientation of $\Sigma(f)-P$$\Sigma(f)-P$.
2. The orientation on $\Sigma(f)-P$$\Sigma(f)-P$ need not extend to

   Tex syntax error

   ${\rm Cl}\Sigma(f)$: take the smooth cone $f:D^3\to\Rr^5$$f:D^3\to\Rr^5$ over a general position map $g:S^2\to\Rr^4$$g:S^2\to\Rr^4$ having only two transverse self-intersection points, where the smooth cone is defined by $f(tx):=(g(x)\sin(\pi t/2),\cos(\pi t/2))$$f(tx):=(g(x)\sin(\pi t/2),\cos(\pi t/2))$, for $x\in S^2$$x\in S^2$ and $t\in[0,1]$$t\in[0,1]$.
3. The orientation on $\Sigma(f)-P$$\Sigma(f)-P$ extends to

   Tex syntax error

   ${\rm Cl}\Sigma(f)$ if $m-n$$m-n$ is odd [Hudson1969, Lemma 11.4].

Remark 7.3 (A canonical orientation on $f\Sigma(f)-fP$$f\Sigma(f)-fP$ for $m-n$$m-n$ even). This remark is added as a complement for Definition 7.1 but is not needed for the definition of the Whitney invariant.

Take a $(2n-m)$$(2n-m)$-base $\xi$$\xi$ at a point $x\in f\Sigma(f)-fP$$x\in f\Sigma(f)-fP$. Since $N$$N$ is oriented, we can take a positive $(m-n)$$(m-n)$-base $\eta_+$$\eta_+$ normal to $f\Sigma(f)$$f\Sigma(f)$ in one sheet of $fN$$fN$. Analogously construct an $(m-n)$$(m-n)$-base $\eta_-$$\eta_-$ for the other sheet of $fN$$fN$. Since $m-n$$m-n$ is even, the orientation of the base $(\xi,\eta_+,\eta_-)$$(\xi,\eta_+,\eta_-)$ of $\Rr^m$$\Rr^m$ does not depend on choosing the first and the other sheet of $fN$$fN$ at $x$$x$. If the base $(\xi,\eta_+,\eta_-)$$(\xi,\eta_+,\eta_-)$ is positive, then call the base $\xi$$\xi$ positive. This is well-defined because a change of the sign of $\xi$$\xi$ forces changes of the signs of $\eta_+,\eta_-$$\eta_+,\eta_-$ and so of $(\xi,\eta_+,\eta_-)$$(\xi,\eta_+,\eta_-)$.

We remark that a change of the orientation of $N$$N$ forces changes of the signs of $\eta_+,\eta_-$$\eta_+,\eta_-$ and so does not change the orientation of $f\Sigma(f)-fP$$f\Sigma(f)-fP$.

8 References

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• [Skopenkov1997] A. Skopenkov, On the deleted product criterion for embeddability of manifolds in $\Rr^m$$\Rr^m$, Comment. Math. Helv. 72 (1997), 543-555.
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• [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
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• [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into $\Rr^{2n-k-1}$$\Rr^{2n-k-1}$, Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
• [Skopenkov2014] A. Skopenkov, How do autodiffeomorphisms act on embeddings, Proc. A of the Royal Society of Edinburgh, 148:4 (2018) 835--848.
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• [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
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• [Skopenkov2016s] A. Skopenkov, Knots, i.e. embeddings of spheres, preprint.
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arxiv:math/0506464 MR2261638 (2007g:57049) Zbl 1113.57013

• [Tonkonog2010] D. Tonkonog, Embedding punctured $n$-manifolds in Euclidean $(2n-1)$-space
• [Vrabec1977] J. Vrabec, Knotting a $k$$k$-connected closed $\PL$$\PL$ $m$$m$-manifold in $E^{2m-k}$$E^{2m-k}$, Trans. Amer. Math. Soc. 233 (1977), 137–165. MR0645405 (58 #31097) Zbl 386.57013
• [Weber1967] C. Weber, Plongements de polyedres dans le domaine metastable, Comment. Math. Helv. 42 (1967), 1-27. MR0238330 (38 #6606) Zbl 0152.22402
• [Yasui1984] T. Yasui, Enumerating embeddings of $n$$n$-manifolds in Euclidean $(2n-1)$$(2n-1)$-space, J. Math. Soc. Japan 36 (1984), no.4, 555–576. MR759414 Zbl 0557.57019

_n:=(1,0,\ldots,0)\in S^n$. == Classification == ;\label{s:e-cla} For the next theorem, [[#The Whitney invariant|the Whitney invariant]] $W$ is defined in $\S$\ref{s:whitney} below. {{beginthm|Theorem}}\label{th4} Assume that $N$ is a closed connected $n$-manifold, and either $n\ge4$ or $n=3$ and we are in the PL category. (a) If $N$ is oriented, the [[#The Whitney invariant|Whitney invariant]], $$W:E^{2n}(N)\to H_1(N;\Zz_{\varepsilon(n-1)}),$$ is a 1-1 correspondence. (b) If $N$ is non-orientable, then there is a 1-1 correspondence $$E^{2n}(N)\to \begin{cases} H_1(N;\Zz_2) & n\text{ is odd}\ \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s & n\text{ is even}\end{cases}.$$ {{endthm}} {{beginthm|Remark|(Comments on the proof)}}\label{r:pth4} Part (a) is proved in \cite[Theorem 2.4]{Haefliger&Hirsch1963} in the smooth category, and in \cite[Theorem 4' in $\S]{Weber1967}, \cite[$\S]{Hudson1969}, \cite[Theorems 1.1 and 1.2]{Vrabec1977} in the PL category, see also \cite[1.3.e]{Haefliger1962b}, \cite{Haefliger1963}, \cite[Theorem 43]{Bausum1975}. Part (b) is proved in \cite[Theorem 43]{Bausum1975} in the smooth category. By \cite[Theorems 1 and 1' in $\S]{Weber1967}, \cite[Theorem 1.1.c]{Skopenkov1997} the proof works also in the PL category. In Part (b) we replaced the kernel $\ker Sq^1$ from \cite[Theorem 43]{Bausum1975} by $\Zz_2^{s-1}$. This is possible because, as a specialist could see, $Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$ is given by multiplication with [[Stiefel-Whitney_characteristic_classes|the first Stefel-Whitney class]] $w_1(N)$ (which equals to the first Wu class $v_1(N)$ \cite[Theorem 11.4]{Milnor&Stasheff1974}). Since $N$ is non-orientable, $w_1(N)\neq 0$. So by Poincaré duality, $\ker Sq^1 \cong \Zz_2^{s-1}$. The 1-1 correspondence from (b) can presumably be defined as a (generalized) Whitney invariant, see \cite{Vrabec1977}, but the proof used 'the Haefliger-Wu invariant' whose definition can be found e.g. in \cite[$\S]{Skopenkov2006}. It would be interesting to check if part (b) is equivalent to different forms of description of $E^{2n}(N)$ \cite[1.3.e]{Haefliger1962b}, \cite{Haefliger1963}, \cite[Theorem 1.1]{Vrabec1977}. {{endthm}} The classification of ''smooth'' [[3-manifolds_in_6-space|embeddings of 3-manifolds in $\Rr^6$]] is more complicated, see Theorem \ref{hicod} below for $l=1$ or \cite{Skopenkov2016t}. Concerning embeddings of connected $n$-manifolds in $\Rr^{2n-1}$ see \cite{Yasui1984} for $n\ge5$, \cite{Skopenkov2016f} [[4-manifolds_in_7-space|for $n=4$]], and \cite{Saeki1999}, \cite{Skopenkov2010}, \cite{Tonkonog2010} for manifolds with boundary. Theorem \ref{th4} is generalized to a description of $E^{2n-k}(N)$ for closed $k$-connected $n$-manifolds $N$, see Theorem \ref{hico}. == Hudson and Lagrangian tori == ; Together with [[3-manifolds_in_6-space#Examples|the Haefliger knotted sphere]] \cite[Example 2.1]{Skopenkov2016t}, \cite[Example 3.4]{Skopenkov2006}, the examples of Hudson tori presented below were the first examples of non-isotopic embeddings in codimension greater than 2. (Hudson's construction \cite{Hudson1963} was not as explicit as those below.) Abbreviate ${\rm i}_{2n,n-1}$ to just ${\rm i}$. {{beginthm|Example}}\label{dh11} Let us construct, for any $a\in\Zz$ and $n\ge2$, a smooth embedding $$\Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.$$ We start with the cases $a=0,1$. Take the standard inclusion $\partial D^2\subset\partial D^{n+1}$. The 'standard embedding' $\Hud_n(0)$ is given by the standard inclusions $$\partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\overset{{\rm i}}\to\Rr^{2n}.$$ Define the `standard embedding' {\rm i}:2D^{n+1}\times S^{n-1}\to\Rr^{2n}$ analogously to ${\rm i}$, where $ means [[Wikipedia:Homothetic_transformation|homothety]] with coefficient 2. Take the embedding $g_1$ given by $$S^n=2\partial D^{n+1}\times1_{n-1}\subset 2D^{n+1}\times S^{n-1}\overset{2{\rm i}}\to\Rr^{2n}.$$ The segment {\rm i}([1,2]1_n\times1_{n-1})$ joins the images of $\Hud_n(0)$ and $g_1$; the interior of this segment misses the images. Let $\Hud_n(1)$ be the linked embedded connected sum of $\Hud_n(0)$ and $g_1$ along this segment, compatible with the orientation, cf. \cite[$\SN is a connected manifold of dimension $n>1$$n>1$, and $m \ge2n+1$$m \ge2n+1$, then every two embeddings $N \to\Rr^m$$N \to\Rr^m$ are isotopic [Skopenkov2016c, Theorem 3.2], [Skopenkov2006, Theorem 2.5]. In this page we summarize the situation for $m=2n\ge6$$m=2n\ge6$ and $N$$N$ is a connected, as well as in some more general situations. For the classification of embeddings of some disconnected manifolds see [Skopenkov2016h].

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$$\S$1, $\S$$\S$3]. Denote $1_n:=(1,0,\ldots,0)\in S^n$$1_n:=(1,0,\ldots,0)\in S^n$.

2 Classification

For the next theorem, the Whitney invariant $W$$W$ is defined in $\S$$\S$5 below.

Theorem 2.1. Assume that $N$$N$ is a closed connected $n$$n$-manifold, and either $n\ge4$$n\ge4$ or $n=3$$n=3$ and we are in the PL category.

(a) If $N$$N$ is oriented, the Whitney invariant,

$\displaystyle W:E^{2n}(N)\to H_1(N;\Zz_{\varepsilon(n-1)}),$ $$\displaystyle W:E^{2n}(N)\to H_1(N;\Zz_{\varepsilon(n-1)}),$$

is a 1-1 correspondence.

(b) If $N$$N$ is non-orientable, then there is a 1-1 correspondence

$\displaystyle E^{2n}(N)\to \begin{cases} H_1(N;\Zz_2) & n\text{ is odd}\\ \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s & n\text{ is even}\end{cases}.$ $$\displaystyle E^{2n}(N)\to \begin{cases} H_1(N;\Zz_2) & n\text{ is odd}\\ \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s & n\text{ is even}\end{cases}.$$

Remark 2.2 (Comments on the proof). Part (a) is proved in [Haefliger&Hirsch1963, Theorem 2.4] in the smooth category, and in [Weber1967, Theorem 4' in $\S$$\S$2], [Hudson1969, $\S$$\S$11], [Vrabec1977, Theorems 1.1 and 1.2] in the PL category, see also [Haefliger1962b, 1.3.e], [Haefliger1963], [Bausum1975, Theorem 43].

Part (b) is proved in [Bausum1975, Theorem 43] in the smooth category. By [Weber1967, Theorems 1 and 1' in $\S$$\S$2], [Skopenkov1997, Theorem 1.1.c] the proof works also in the PL category.

In Part (b) we replaced the kernel $\ker Sq^1$$\ker Sq^1$ from [Bausum1975, Theorem 43] by $\Zz_2^{s-1}$$\Zz_2^{s-1}$. This is possible because, as a specialist could see, $Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$$Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$ is given by multiplication with the first Stefel-Whitney class $w_1(N)$$w_1(N)$ (which equals to the first Wu class $v_1(N)$$v_1(N)$ [Milnor&Stasheff1974, Theorem 11.4]). Since $N$$N$ is non-orientable, $w_1(N)\neq 0$$w_1(N)\neq 0$. So by Poincaré duality, $\ker Sq^1 \cong \Zz_2^{s-1}$$\ker Sq^1 \cong \Zz_2^{s-1}$.

The 1-1 correspondence from (b) can presumably be defined as a (generalized) Whitney invariant, see [Vrabec1977], but the proof used 'the Haefliger-Wu invariant' whose definition can be found e.g. in [Skopenkov2006, $\S$$\S$5]. It would be interesting to check if part (b) is equivalent to different forms of description of $E^{2n}(N)$$E^{2n}(N)$ [Haefliger1962b, 1.3.e], [Haefliger1963], [Vrabec1977, Theorem 1.1].

The classification of smooth embeddings of 3-manifolds in $\Rr^6$$\Rr^6$ is more complicated, see Theorem 6.3 below for $l=1$$l=1$ or [Skopenkov2016t].

Concerning embeddings of connected $n$$n$-manifolds in $\Rr^{2n-1}$$\Rr^{2n-1}$ see [Yasui1984] for $n\ge5$$n\ge5$, [Skopenkov2016f] for $n=4$$n=4$, and [Saeki1999], [Skopenkov2010], [Tonkonog2010] for manifolds with boundary.

Theorem 2.1 is generalized to a description of $E^{2n-k}(N)$$E^{2n-k}(N)$ for closed $k$$k$-connected $n$$n$-manifolds $N$$N$, see Theorem 6.2.

3 Hudson and Lagrangian tori

Together with the Haefliger knotted sphere [Skopenkov2016t, Example 2.1], [Skopenkov2006, Example 3.4], the examples of Hudson tori presented below were the first examples of non-isotopic embeddings in codimension greater than 2. (Hudson's construction [Hudson1963] was not as explicit as those below.) Abbreviate

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${\rm i}_{2n,n-1}$ to just

Tex syntax error

${\rm i}$.

Example 3.1. Let us construct, for any $a\in\Zz$$a\in\Zz$ and $n\ge2$$n\ge2$, a smooth embedding

$\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.$ $$\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.$$

We start with the cases $a=0,1$$a=0,1$.

Take the standard inclusion $\partial D^2\subset\partial D^{n+1}$$\partial D^2\subset\partial D^{n+1}$. The 'standard embedding' $\Hud_n(0)$$\Hud_n(0)$ is given by the standard inclusions

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$$\displaystyle \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\overset{{\rm i}}\to\Rr^{2n}.$$

Define the `standard embedding'

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$2{\rm i}:2D^{n+1}\times S^{n-1}\to\Rr^{2n}$ analogously to

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${\rm i}$, where $2$$2$ means homothety with coefficient 2.

Take the embedding $g_1$$g_1$ given by

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$$\displaystyle S^n=2\partial D^{n+1}\times1_{n-1}\subset 2D^{n+1}\times S^{n-1}\overset{2{\rm i}}\to\Rr^{2n}.$$

The segment

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$2{\rm i}([1,2]1_n\times1_{n-1})$ joins the images of $\Hud_n(0)$$\Hud_n(0)$ and $g_1$$g_1$; the interior of this segment misses the images. Let $\Hud_n(1)$$\Hud_n(1)$ be the linked embedded connected sum of $\Hud_n(0)$$\Hud_n(0)$ and $g_1$$g_1$ along this segment, compatible with the orientation, cf. [Avvakumov2017, $\S$$\S$1.5]. (Here 'linked' means that the images of the embeddings are not contained in disjoint cubes, unlike for the unlinked embedded connected sum [Skopenkov2016c, $\S$$\S$5].)

For $a\in\Zz$$a\in\Zz$ we repeat the above construction of $g_1$$g_1$ replacing $2\partial D^{n+1}\times1_{n-1}$$2\partial D^{n+1}\times1_{n-1}$ by $|a|$$|a|$ copies $(1+\frac1k)\partial D^{n+1}\times1_{n-1}$$(1+\frac1k)\partial D^{n+1}\times1_{n-1}$ of $S^n$$S^n$, $k=1,\ldots,|a|$$k=1,\ldots,|a|$. The copies are outside $D^{n+1}\times S^{n-1}$$D^{n+1}\times S^{n-1}$ and are `parallel' to $\partial D^{n+1}\times1_{n-1}$$\partial D^{n+1}\times1_{n-1}$. The copies have the standard orientation for $a>0$$a>0$ or the opposite orientation for $a<0$$a<0$. Then we make embedded connected sum along natural segments joining every $k$$k$-th copy to the $(k+1)$$(k+1)$-th copy. We obtain an embedding $g_a:S^n\to\Rr^{2n}$$g_a:S^n\to\Rr^{2n}$ which has disjoint images with $\Hud_n(0)$$\Hud_n(0)$. Let $\Hud_n(a)$$\Hud_n(a)$ be the linked embedded connected sum of $\Hud_n(0)$$\Hud_n(0)$ and $g_a$$g_a$.

The original motivation for Hudson was that $\Hud_n(1)$$\Hud_n(1)$ is not isotopic to $\Hud_n(0)$$\Hud_n(0)$ for any $n\ge3$$n\ge3$ (this is a particular case of Proposition 3.2 below). One might guess that $\Hud_n(a)$$\Hud_n(a)$ is not isotopic to $\Hud_n(a')$$\Hud_n(a')$ for $a\ne a'$$a\ne a'$ and that a $\Zz$$\Zz$-valued invariant of $E^{2n}(S^1 \times S^{n-1})$$E^{2n}(S^1 \times S^{n-1})$ can be defined by the homotopy class of the map

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$$\displaystyle S^n\overset g\to S^{2n}-{\rm i}(D^{n+1}\times S^{n-1})\sim S^{2n}-S^{n-1}\sim S^n \quad\text{which is}\quad a\in\pi_n(S^n)\cong\Zz.$$

However, this is only true for $n$$n$ odd.

Proposition 3.2. For $n\ge3$$n\ge3$ odd $\Hud_n(a)$$\Hud_n(a)$ is isotopic to $\Hud_n(a')$$\Hud_n(a')$ if and only if $a=a'$$a=a'$.

For $n\ge4$$n\ge4$ even $\Hud_n(a)$$\Hud_n(a)$ is isotopic to $\Hud_n(a')$$\Hud_n(a')$ if and only if $a\equiv a'\text{ mod}2$$a\equiv a'\text{ mod}2$.

Proposition 3.2 follows by calculation of the Whitney invariant (Remark 5.3.d below) and, for $n$$n$ even, by Theorem 2.1. This proposition holds with the same proof in the piecewise smooth category, whose definition is recalled in [Skopenkov2016f, Remark 1.1]). Proposition 3.2 also holds in the PL category (with an analogous construction of $\Hud_n(a)$$\Hud_n(a)$ for the PL category). It would be interesting to find an explicit construction of an isotopy between $\Hud_{2k}(a)$$\Hud_{2k}(a)$ and $\Hud_{2k}(a+2)$$\Hud_{2k}(a+2)$, cf. [Vrabec1977, $\S$$\S$5]. Analogously, $\Hud_2(a)$$\Hud_2(a)$ is not isotopic to $\Hud_2(a')$$\Hud_2(a')$ if $a\not\equiv a'\text{ mod}2$$a\not\equiv a'\text{ mod}2$. It would be interesting to know if the converse holds, e.g. is $\Hud_2(0)$$\Hud_2(0)$ (PS or smoothly) isotopic to $\Hud_2(2)$$\Hud_2(2)$.

Example 3.3. Take any $a\in\Zz$$a\in\Zz$. Take a map $\overline a:S^{n-1}\to S^{n-1}$$\overline a:S^{n-1}\to S^{n-1}$ of degree $a$$a$ (so we can take $\overline 1=\id$$\overline 1=\id$). Recall that $D^{n+1}=\{(y,x)\in D^n\times D^1\ :\ |y|^2+|x|^2\le1\}$$D^{n+1}=\{(y,x)\in D^n\times D^1\ :\ |y|^2+|x|^2\le1\}$. Define the smooth embedding $\Hud_n'(a)$$\Hud_n'(a)$ to be the composition

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$$\displaystyle S^1\times S^{n-1}\overset{\widehat a}\to D^{n+1}\times S^{n-1}\overset{\rm i}\to\Rr^{2n} \quad\text{where}\quad \widehat a(e^{i\theta},y):=(\overline a(y)\cos\theta,\sin\theta,y).$$

Figure 1: The Hudson torus

Let us present a geometric description of this embedding. Define a map $\widetilde a:S^0\times S^{n-1}\to S^{n-1}$$\widetilde a:S^0\times S^{n-1}\to S^{n-1}$ by $\widetilde a(s,t):=s\overline a(t)$$\widetilde a(s,t):=s\overline a(t)$. This map gives an embedding

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$$\displaystyle S^0\times S^{n-1}\xrightarrow{\widetilde a\times {\rm pr}_2}S^{n-1}\times S^{n-1}\subset D^{n+1}\times S^{n-1}\overset{{\rm i}}\to\Rr^{2n}.$$

See Figure 1. The image of

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$\widetilde a\times{\rm pr}_2$ is the union of the graphs of the maps $\overline a$$\overline a$ and $-\overline a$$-\overline a$. For any $t\in S^{n-1}$$t\in S^{n-1}$ the disk

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${\rm i}(D^{n+1}\times t)$ intersects the image at two points lying in

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${\rm i}(D^n\times t)$, i.e., at the image of an embedding

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$S^0\times t\to {\rm i}(D^n\times t)$. The embedding $\Hud_n'(a)$$\Hud_n'(a)$ is obtained by extending the latter embeddings to embeddings

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$S^1\times t\to {\rm i}(D^{n+1}\times t)$ for all $t$$t$. See Figure 2.

Figure 2: To the construction of the Hudson torus

Remark 3.4. (a) The analogue of Proposition 3.2 for $\Hud_n$$\Hud_n$ replaced to $\Hud_n'$$\Hud_n'$ holds, with an analogous proof.

(b) The embeddings $\Hud_n(a)$$\Hud_n(a)$ and $\Hud_n'(a)$$\Hud_n'(a)$ are smoothly isotopic for $n\ge4$$n\ge4$ and are PS isotopic for $n\ge3$$n\ge3$ [Skopenkov2006a, commutativity of the left upper square in the Restriction Lemma 5.2], [Skopenkov2015a, Lemma 2.15.c] (see [Skopenkov2016f, Remark 1.2]). This follows by calculation of the Whitney invariant (Remark 5.3.d below). It would be interesting to know if they are smoothly isotopic for $n=3$$n=3$. It would be interesting to know if they are piecewise smoothly isotopic for $n=2$$n=2$.

(b') Although construction of $\Hud_n(a)$$\Hud_n(a)$ and $\Hud_n'(a)$$\Hud_n'(a)$ is explicit, no explicit construction of an isotopy between them is known. Proof from (b) of their being isotopic is by calculating invariants, and using completeness result for the invariants. This proof is fairly constructive, so in principle it does give a specific isotopy. However, the construction is so complicated that one would not call such an isotopy explicit. This is analogous to the Smale `turning sphere inside out' theorem (although the Smale theorem concerns regular homotopy of immersions not isotopy of embeddings).

In higher dimensions, a proof that any embedding is isotopic to some of the constructed embeddings is even less likely to use an explicit construction. The only proofs we have go through calculating the invariants and using classification results.

(c) For $n=2$$n=2$ Example 3.3 gives what we call the 'left' Hudson torus. The 'right' Hudson torus is constructed analogously. It is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of $S^1\times S^1$$S^1\times S^1$. The right and the left Hudson tori are not isotopic by Remark 5.3.d below.

(d) Analogously one constructs the Hudson torus $\Hud_{n,p}(a):S^p\times S^{n-p}\to\Rr^{2n-p+1}$$\Hud_{n,p}(a):S^p\times S^{n-p}\to\Rr^{2n-p+1}$ for $a\in\Zz$$a\in\Zz$ and $n>p\ge0$$n>p\ge0$ or, more generally, $\Hud_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$$\Hud_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$ for $a\in\pi_n(S^{m-n+p-1})$$a\in\pi_n(S^{m-n+p-1})$ and $m>n>p\ge0$$m>n>p\ge0$. There are versions $\Hud'_{m,n,p}(a)$$\Hud'_{m,n,p}(a)$ of these constructions corresponding to Definition 3.3. For $p=0$$p=0$ this corresponds to the Zeeman map [Skopenkov2016h, Definition 2.2] and its composition with 'the unframed second Kirby move' [Skopenkov2015a, $\S$$\S$2.3]. It would be interesting to know if the links $\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$$\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$ are isotopic, cf. [Skopenkov2015a, Remark 2.7.b]. These constructions can be further generalized [Skopenkov2016k].

Example 3.5. Define a smooth embedding

$\displaystyle l:S^1\times S^{n-1}\to \C^n=\R^{2n}\quad\text{by}\quad l(z,x) := \left(1+\frac z2\right)x.$ $$\displaystyle l:S^1\times S^{n-1}\to \C^n=\R^{2n}\quad\text{by}\quad l(z,x) := \left(1+\frac z2\right)x.$$

Here $S^1\subset\R^2=\C$$S^1\subset\R^2=\C$ and $S^{n-1}\subset\R^n\subset\C^n$$S^{n-1}\subset\R^n\subset\C^n$. This embedding is known to be Lagrangian; see more in [Nemirovski2024].

4 Action by linked embedded connected sum

In this section we generalize the construction of the Hudson torus $\Hud(a)$$\Hud(a)$. Let $N$$N$ be a closed connected oriented $n$$n$-manifold. We work in the smooth category which we omit. Apparently analogous results hold for $n\ge3$$n\ge3$ in the PL and PS categories (see [Skopenkov2016f, Remark 1.2]).

Example 4.1. For any $n\ge4$$n\ge4$, an embedding $f_0:N\to\Rr^{2n}$$f_0:N\to\Rr^{2n}$ and $a\in H_1(N;\Zz)$$a\in H_1(N;\Zz)$, we shall construct an embedding $f_a:N\to\Rr^{2n}$$f_a:N\to\Rr^{2n}$. This embedding is said to be obtained by linked embedded connected sum of $f_0$$f_0$ with an $n$$n$-sphere representing the `homology Alexander dual' $A:=\widehat{A_{f_0}}a\in H_n(C_{f_0})$$A:=\widehat{A_{f_0}}a\in H_n(C_{f_0})$ of $a$$a$ (defined in [Skopenkov2005, Alexander Duality Lemma 4.6]).

Represent $a$$a$ by an embedding $a:S^1\to N$$a:S^1\to N$. By definition, the class $A$$A$ is represented by properly oriented $\nu_{f_0}^{-1}a(S^1)$$\nu_{f_0}^{-1}a(S^1)$. Since any orientable bundle over $S^1$$S^1$ is trivial, $\nu_{f_0}^{-1}a(S^1)\cong S^1\times S^{n-1}$$\nu_{f_0}^{-1}a(S^1)\cong S^1\times S^{n-1}$. Take an embedding $g:S^1\times S^{n-1}\to C_{f_0}$$g:S^1\times S^{n-1}\to C_{f_0}$ whose image is $\nu_{f_0}^{-1}a(S^1)$$\nu_{f_0}^{-1}a(S^1)$ and which represents $A$$A$. By embedded surgery on $S^1\times1_{n-1}\subset S^1\times S^{n-1}$$S^1\times1_{n-1}\subset S^1\times S^{n-1}$ we obtain an embedding $g_1:S^n\to C_{f_0}$$g_1:S^n\to C_{f_0}$ representing $A$$A$ (see details in Proposition 4.2 below). Define $f_a$$f_a$ to be the linked embedded connected sum of $f_0$$f_0$ and $g_1$$g_1$, along some arc joining their images.

Proposition 4.2 (Embedded surgery). For any $n\ge3$$n\ge3$, a neighborhood $U$$U$ of a codimension at least 3 subpolyhedron in $\Rr^{2n}$$\Rr^{2n}$ and an embedding $g:S^1\times S^{n-1}\to\Rr^{2n}-U$$g:S^1\times S^{n-1}\to\Rr^{2n}-U$ there is an embedding $g_1:S^n\to\Rr^{2n}-U$$g_1:S^n\to\Rr^{2n}-U$ homologous to $g$$g$.

Proof. Take a vector field on $g(S^1\times1_{n-1})$$g(S^1\times1_{n-1})$ normal to $g(S^1\times S^{n-1})$$g(S^1\times S^{n-1})$. Extend $g|_{S^1\times1_{n-1}}$$g|_{S^1\times1_{n-1}}$ along this vector field to a map $\overline b:D^2\to\Rr^{2n}$$\overline b:D^2\to\Rr^{2n}$.

Since $2n>4$$2n>4$ and $U$$U$ is a neighborhood $U$$U$ of a codimension at least 3 subpolyhedron, by general position we may assume that $\overline b$$\overline b$ is an embedding and that

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$\overline b({\rm Int}D^2)$ misses $U\cup g(S^1\times S^{n-1})$$U\cup g(S^1\times S^{n-1})$.

Since $n-1>1$$n-1>1$, we have $\pi_1(V_{2n-2,n-1})=0$$\pi_1(V_{2n-2,n-1})=0$. Hence the standard $(n-1)$$(n-1)$-framing of $S^1\times1_{n-1}$$S^1\times1_{n-1}$ in $S^1\times S^{n-1}$$S^1\times S^{n-1}$ extends to an $(n-1)$$(n-1)$-framing on $\overline b(D^2)$$\overline b(D^2)$ in $\Rr^{2n}$$\Rr^{2n}$. Thus $\overline b$$\overline b$ extends to an embedding

$\displaystyle \widehat b:D^2\times D^{n-1}\to\R^{2n}-U\quad\text{such that}\quad \widehat b(\partial D^2\times D^{n-1})\subset g(S^1\times S^{n-1}).$ $$\displaystyle \widehat b:D^2\times D^{n-1}\to\R^{2n}-U\quad\text{such that}\quad \widehat b(\partial D^2\times D^{n-1})\subset g(S^1\times S^{n-1}).$$

Take an embedding $g_1:S^n\to\R^{2n}-U$$g_1:S^n\to\R^{2n}-U$ such that

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$$\displaystyle g_1(S^n)\ =\ g(S^1\times S^{n-1})-\widehat b(\partial D^2\times{\rm Int}D^{n-1}) \bigcup\limits_{\widehat b(\partial D^2\times\partial D^{n-1})} \widehat b(D^2\times\partial D^{n-1})\ \cong\ S^n.$$

with proper orientation so that $g_1$$g_1$ is homologous to $g$$g$. QED

The isotopy class of the embedding $f_a$$f_a$ is independent of the choises in the construction. The independence of the arc and of the maps $g,g_1$$g,g_1$ follows by $n\ge3$$n\ge3$ and by Proposition 4.3 below, respectively.

By Definition 5.1 of the Whitney invariant, $W(f_a,f_0)$$W(f_a,f_0)$ is $a$$a$ for $n\ge3$$n\ge3$ odd and $a \mod2$$a \mod2$ for $n\ge4$$n\ge4$ even. Thus by Theorem 2.1.a for $n\ge4$$n\ge4$ all isotopy classes of embeddings $N\to\Rr^{2n}$$N\to\Rr^{2n}$ can be obtained from any chosen embedding $f_0$$f_0$ by the above construction.

Proposition 4.3. For any $n\ge4$$n\ge4$ both the linked embedded connected sum and parametric connected sum (introduced in [Skopenkov2006a], [Skopenkov2015a]) define free transitive actions of $H_1(N;\Zz_{\varepsilon(n-1)})$$H_1(N;\Zz_{\varepsilon(n-1)})$ on $E^{2n}(N)$$E^{2n}(N)$.

This follows by Theorem 2.1.a and by [Skopenkov2014, Remark 18.a].

5 The Whitney invariant

Let $N$$N$ be a closed $n$$n$-manifold. Take an embedding $f_0:N\to\Rr^m$$f_0:N\to\Rr^m$. Fix an orientation on $\Rr^m$$\Rr^m$. For any other embedding $f \colon N \to \Rr^m$$f \colon N \to \Rr^m$ we define the Whitney invariant

$\displaystyle W(f, f_0)=W_{f_0}(f)=W(f)\in H_{2n-m+1}(N;\Zz_N).$ $$\displaystyle W(f, f_0)=W_{f_0}(f)=W(f)\in H_{2n-m+1}(N;\Zz_N).$$

Here the coefficients $\Zz_N$$\Zz_N$ are $\Zz$$\Zz$ if $N$$N$ is oriented and $m-n$$m-n$ is odd, and are $\Zz_2$$\Zz_2$ otherwise.

Roughly speaking,

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$W(f):=[{\rm Cl}\Sigma(H)]$ is defined as the homology class of the closure of the self-intersection set of a general position homotopy $H$$H$ between $f$$f$ and $f_0$$f_0$.

This is formalized in Definition 5.2 in the smooth category, following [Skopenkov2010], see also [Haefliger&Hirsch1963]. The definition in the PL category is analogous [Hudson1969, $\S$$\S$11], [Vrabec1977, p. 145], [Skopenkov2006, $\S$$\S$2.4 `The Whitney invariant']. We begin by presenting a simpler definition, Definition 5.1, for a particular case.

For Theorem 2.1 only the case $m=2n$$m=2n$ is required.

Definition 5.1. Assume that $N$$N$ is $(2n-m)$$(2n-m)$-connected and $2m\ge3n+3$$2m\ge3n+3$. Then by [Haefliger&Hirsch1963, Theorem 3.1.b] restrictions of $f$$f$ and $f_0$$f_0$ to $N_0$$N_0$ are isotopic, cf. [Takase2006, Lemma 2.2]. (Here is sketch of an argument. Using the Smale-Hirsch classification of immersions we obtain that restrictions of $f$$f$ and $f_0$$f_0$ to $N_0$$N_0$ are `regular homotopic', see [Koschorke2013, Definition 2.7]. Since $N$$N$ is $(2n-m)$$(2n-m)$-connected, $N_0$$N_0$ retracts to an $(m-n-1)$$(m-n-1)$-dimensional polyhedron. Therefore these restrictions are isotopic.)

So we can make an isotopy of $f$$f$ and assume that $f=f_0$$f=f_0$ on $N_0$$N_0$. Take a general position homotopy $F:B^n\times I\to\Rr^m$$F:B^n\times I\to\Rr^m$ relative to $\partial B^n$$\partial B^n$ between the restrictions of $f$$f$ and $f_0$$f_0$ to $B^n$$B^n$. Let $f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$$f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$ (`the intersection of this homotopy with $f(N-B^n)$$f(N-B^n)$').

Since $n+2(n+1)<2m$$n+2(n+1)<2m$, by general position

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${\rm Cl}(f\cap F)$ is a compact $(2n+1-m)$$(2n+1-m)$-manifold whose boundary is contained in $\partial N_0$$\partial N_0$.

So $f\cap F$$f\cap F$ carries a homology class with $\Zz_2$$\Zz_2$ coefficients. If $m-n$$m-n$ is odd and $N$$N$ is oriented, then $f\cap F$$f\cap F$ has a natural orientation defined below, and so carries a homology class with $\Zz$$\Zz$ coefficients. Define $W(f)$$W(f)$ to be the homology class:

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$$\displaystyle W(f):=[{\rm Cl}(f\cap F)]\in H_{2n-m+1}(N_0,\partial N_0;\Zz_N)\cong H_{2n-m+1}(N;\Zz_N).$$

The orientation on $f\cap F$$f\cap F$ (extendable to

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${\rm Cl}(f\cap F)$) is defined (for $m-n$$m-n$ odd and $N$$N$ is oriented) as follows (cf. Remark 7.3). For any point $x_f\in f\cap F$$x_f\in f\cap F$ take a base $\xi_f$$\xi_f$ at $x_f$$x_f$ tangent to $f\cap F$$f\cap F$. Complete this base to a positive base $(\xi_f,\eta_f)$$(\xi_f,\eta_f)$ tangent to $N$$N$. Since $n+2(n+1)<2m$$n+2(n+1)<2m$, by general position there is a unique point $x_F\in B^n\times I$$x_F\in B^n\times I$ such that $Fx_F=fx_f$$Fx_F=fx_f$. The tangent base $\xi_f$$\xi_f$ at $x_f$$x_f$ thus gives a base $\xi_F$$\xi_F$ at $x_F$$x_F$ tangent to $B^n\times I$$B^n\times I$ such that $df(x_f)\xi_f=dF(x_F)\xi_F$$df(x_f)\xi_f=dF(x_F)\xi_F$. Complete this base $\xi_F$$\xi_F$ to a positive base $(\xi_F,\eta_F)$$(\xi_F,\eta_F)$ tangent to $B^n\times I$$B^n\times I$, where the orientation on $B^n$$B^n$ comes from $N$$N$. The union $\zeta=(df(x_f)\xi_f,df(x_f)\eta_f,dF(x_F)\eta_F)$$\zeta=(df(x_f)\xi_f,df(x_f)\eta_f,dF(x_F)\eta_F)$ of the images of the constructed bases is a base at $fx_f=Fx_F$$fx_f=Fx_F$ of $\Rr^m$$\Rr^m$. If $\zeta$$\zeta$ is positive, then call the tangent base $\xi_f$$\xi_f$ of $f\cap F$$f\cap F$ `positive'. Since a change of the orientation on $f\cap F$$f\cap F$ forces a change of the orientation of $\zeta$$\zeta$, this condition indeed defines an orientation on $f\cap F$$f\cap F$.

Definition 5.2. Assume that $m\ge n+2$$m\ge n+2$. Take a general position homotopy $H:N\times I\to\Rr^m\times I$$H:N\times I\to\Rr^m\times I$ between $f_0$$f_0$ and $f$$f$.

The closure

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${\rm Cl}\Sigma(H)$ of the self-intersection set carries a cycle mod 2. If $N$$N$ is oriented and $m-n$$m-n$ is odd, the closure also carries an integer cycle. See [Hudson1967, $\S$$\S$11], [Skopenkov2006, $\S$$\S$2.3 `The Whitney obstruction']. Let us informally explain these facts. For $2m\ge3n+2$$2m\ge3n+2$ by general position the closure

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${\rm Cl}\Sigma(H)$ can be assumed to be a submanifold. In general, since $m\ge n+2$$m\ge n+2$, by general position the closure has codimension 2 singularities, as defined in $\S$$\S$7. So the closure carries a cycle mod 2. When $m-n$$m-n$ is odd the closure also has a canonical orientation (see Definition 7.1 and Remark 7.2), so the closure carries an integer cycle.

Define the Whitney invariant to be the homology class:

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$$\displaystyle W(f):=[{\rm Cl}\Sigma(H)]\in H_{2n-m+1}(N\times I;\Zz_N)\cong H_{2n-m+1}(N;\Zz_N).$$

Clearly, $W(f) = W(f')$$W(f) = W(f')$ if $f$$f$ is isotopic to $f'$$f'$. Hence the Whitney invariant defines a map

$\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_N),\quad [f] \mapsto W(f).$ $$\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_N),\quad [f] \mapsto W(f).$$

Clearly, $W(f_0)=0$$W(f_0)=0$ (for both definitions).

The definition of $W$$W$ depends on the choice of $f_0$$f_0$, but we write $W$$W$ not $W_{f_0}$$W_{f_0}$ for brevity.

Remark 5.3. (a) The Whitney invariant is well-defined by Definition 5.2, i.e. is independent of the choice of a general position homotopy $H:N\times I\to\Rr^m\times I$$H:N\times I\to\Rr^m\times I$ from $f_0$$f_0$ to $f$$f$.

This follows from the equality

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$[{\rm Cl}\Sigma(H_0)]−[{\rm Cl}\Sigma(H_1)] = \partial [{\rm Cl}\Sigma(H_{01})]$ for a general position homotopy $H_{01}:N\times I\times I\to\Rr^m\times I\times I$$H_{01}:N\times I\times I\to\Rr^m\times I\times I$ between general position homotopies $H_0,H_1:N\times I\to\Rr^m\times I$$H_0,H_1:N\times I\to\Rr^m\times I$ from $f_0$$f_0$ to $f$$f$. See details in [Hudson1969, $\S$$\S$11].

(b) Definition 5.1 is a particular case of Definition 5.2. (Indeed, if $f=f_0$$f=f_0$ on $N_0$$N_0$, we can take $H$$H$ to be fixed on $N_0$$N_0$. See details in [Skopenkov2010, Difference Lemma 2.4].) Hence the Whitney invariant is well-defined by Definition 5.1, i.e. independent of the choice of $F$$F$ and of the isotopy making $f=f_0$$f=f_0$ outside $B^n$$B^n$.

(c) The class $W(f)$$W(f)$ is independent of the choice of the orientation on $N$$N$ (because a change of the orientation on $N$$N$ forces a change of the orientation on $f\cap F$$f\cap F$ or on

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${\rm Cl}\Sigma(H)$). For the reflection $\sigma:\Rr^m\to\Rr^m$$\sigma:\Rr^m\to\Rr^m$ with respect to a hyperplane we have $W(\sigma\circ f)=-W(f)$$W(\sigma\circ f)=-W(f)$ (because a change of the orientation on $\Rr^m$$\Rr^m$ forces a change of the orientation on $f\cap F$$f\cap F$ or on

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${\rm Cl}\Sigma(H)$; for Definition 5.1 also observe that we may assume that $f=f_0=\sigma\circ f$$f=f_0=\sigma\circ f$ on $N_0$$N_0$).

(d) For the Hudson tori $W(\Hud_n(a))=W(\Hud'_n(a))$$W(\Hud_n(a))=W(\Hud'_n(a))$ is $a$$a$ or $a\mod2$$a\mod2$ for $n\ge3$$n\ge3$, and $W(\Hud'_2(a))=(a\mod2,0)$$W(\Hud'_2(a))=(a\mod2,0)$.

For $\Hud_n(a)$$\Hud_n(a)$ this is clear by Definition 5.1. For $\Hud'_n(a)$$\Hud'_n(a)$ and $n\ge3$$n\ge3$ this was proved in [Hudson1963] (with a different but equivalent definition of the Whitney invariant; using and proving a particular case of Remark 5.3.f). For $\Hud'_2(a)$$\Hud'_2(a)$ the proof is analogous.

(e) $W(f\#g)=W(f)$$W(f\#g)=W(f)$ for any pair of embeddings $f:N\to\Rr^m$$f:N\to\Rr^m$ and $g:S^n\to\Rr^m$$g:S^n\to\Rr^m$. This is clear by Definition 5.1 because $W(f\#g)-W(f)=W(f\#g,f_0)-W(f,f_0)=W(f\#g,f)=0$$W(f\#g)-W(f)=W(f\#g,f_0)-W(f,f_0)=W(f\#g,f)=0$. Let us prove the latter equality. Take the identical isotopy $H_f$$H_f$ of $f$$f$ and a general position homotopy $H_g$$H_g$ between $g$$g$ and the standard embedding. Then the boundary connected sum $H_f\sharp H_g$$H_f\sharp H_g$ is a general position homotopy between $f\#g$$f\#g$ and an embedding isotopic to $f$$f$. The cycle

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${\rm Cl}\Sigma(H_f\sharp H_g)={\rm Cl}\Sigma H_g$ is null-homologous in $S^n$$S^n$ and hence in $N\cong N\#S^n$$N\cong N\#S^n$; cf. [Skopenkov2008, Addendum to the Classification Theorem].

(f) For $m=2n+1$$m=2n+1$ and $N=S^n\sqcup S^n$$N=S^n\sqcup S^n$ the Whitney invariant equals to the pair of linking coefficients [Skopenkov2016h, $\S$$\S$3].

(g) The Whitney invariant need not be a bijection for $m<2n$$m<2n$. This is seen, for example, by applying Theorem 6.4 below in case of knotted tori [Skopenkov2016k, Theorem 5.1]) or by taking $n$$n$ even, $N$$N$ non-orientable, $m=2n$$m=2n$ and applying by Theorem 2.1.b.

(h) The paper [Haefliger&Hirsch1963, $\S$$\S$2] gives yet another equivalent definition of the Whitney invariant, and proves its injectivity under certain restrictions (the Haefliger-Hirsch $h$$h$-principle, see Theorem 6.2).

(i) Let $N$$N$ be a closed orientable $n$$n$-manifold. For two embeddings $f,g:N\to\R^{2n-1}$$f,g:N\to\R^{2n-1}$ the invariant defined in [Haefliger&Hirsch1963, $\S$$\S$2] equals plus or minus their Whitney invariant $W(f,g)$$W(f,g)$ from Definition 5.1. This is proved analogously to the following version.

Let $D^n$$D^n$ and $B^n$$B^n$ be closed $n$$n$-balls in $N$$N$ such that

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$D^n\subset{\rm Int}B^n$. Let $f:N\to\R^{2n-1}$$f:N\to\R^{2n-1}$ a general position immersion which is an embedding on the closure of $N-D^n$$N-D^n$. Let $N_0 := N-B^n$$N_0 := N-B^n$ and $f_0:=f|_{N-B^n}$$f_0:=f|_{N-B^n}$. We omit $\Z_{(n)}$$\Z_{(n)}$-coefficients. Then $f_0^{-1}f(D^n)$$f_0^{-1}f(D^n)$ represents a 1-cycle in $N$$N$. The homology class of this 1-cycle equals minus the normal Stiefel-Whitney class $PD\overline{W}^{n-1}(N)\in H_1(N)$$PD\overline{W}^{n-1}(N)\in H_1(N)$. This follows because

$\displaystyle 0 \overset{(1)}= [f^{-1}f'(N)] \overset{(2)}= [f_0^{-1}f'(N_0)]+[f_0^{-1}f'(B^n)] \overset{(3)} = PD\overline{W}^{n-1}(N)+[f_0^{-1}f(D^n)].$ $$\displaystyle 0 \overset{(1)}= [f^{-1}f'(N)] \overset{(2)}= [f_0^{-1}f'(N_0)]+[f_0^{-1}f'(B^n)] \overset{(3)} = PD\overline{W}^{n-1}(N)+[f_0^{-1}f(D^n)].$$

Here

• $f':N\to\R^{2n-1}$$f':N\to\R^{2n-1}$ is the map formed by a general position small length vector field normal to $f$$f$, and non-zero over $f(B^n)$$f(B^n)$;

• equality (1) holds because $f'$$f'$ is homotopic to a map whose image is disjoint from $f$$f$;

• equality (2) holds because $f_0$$f_0$-preimages of both $f'(N_0)$$f'(N_0)$ and $f'B^n$$f'B^n$ are 1-cycles, and $f|_{B^n}$$f|_{B^n}$-preimage of $f'(N)$$f'(N)$ is a 1-cycle which is null-homologous;

• equality (3) holds because the Euler class of a bundle is dual to the homology class of zeroes of a general position section, and because $[f_0^{-1}f'(B^n)] = [f_0^{-1}f'(D^n)] \overset{(4)}= [f_0^{-1}f(D^n)]$$[f_0^{-1}f'(B^n)] = [f_0^{-1}f'(D^n)] \overset{(4)}= [f_0^{-1}f(D^n)]$;

• equality (4) holds because $f(D^n)$$f(D^n)$ is homologous relative to the boundary and outside $f(N_0)$$f(N_0)$ to the union of $f'(D^n)$$f'(D^n)$ and the the normal vectors at points of $f(\partial D^n)$$f(\partial D^n)$.

6 A generalization to highly-connected manifolds

In this section let $N$$N$ be a closed orientable homologically $k$$k$-connected $n$$n$-manifold, $k\ge0$$k\ge0$. Recall the unknotting theorem [Skopenkov2016c, Theorem 2.4] that all embeddings $N \to\Rr^m$$N \to\Rr^m$ are isotopic when $m\ge 2n-k+1$$m\ge 2n-k+1$ and $n\ge2k+2$$n\ge2k+2$. In this section we generalize Theorem 2.1 to a description of $E^{2n-k}(N)$$E^{2n-k}(N)$ and further to $E^m(N)$$E^m(N)$ for $m\ge2n-2k+1$$m\ge2n-2k+1$.

6.1 Examples

Some simple examples are the Hudson tori $\Hud_{m,n,p}(a), \Hud'_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$$\Hud_{m,n,p}(a), \Hud'_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$.

Example 6.1 (cf. [Skopenkov2010, Definition 1.4]). Assume that $N$$N$ is $k$$k$-connected and $n-k\ge3$$n-k\ge3$. Then for an embedding $f_0:N\to\Rr^{2n-k}$$f_0:N\to\Rr^{2n-k}$ and a class $a\in H_{k+1}(N;\Zz)$$a\in H_{k+1}(N;\Zz)$ one can construct an embedding $f_a:N\to\Rr^{2n-k}$$f_a:N\to\Rr^{2n-k}$ by linked embedded connected sum analogously to the case $k=0$$k=0$ presented in Example 4.1.

We have $W(f_a,f_0)=a$$W(f_a,f_0)=a$ for the Whitney invariant. Hence by Theorem 6.2 below this construction gives a free transitive action of $H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$$H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$ on $E^{2n-k}(N)$$E^{2n-k}(N)$, provided $n\ge k+3$$n\ge k+3$ or $n\ge2k+4$$n\ge2k+4$ in the PL or smooth categories, respectively.

The embedding $f_a$$f_a$ has an alternative construction using parametric connected sum [Skopenkov2014, Remark 18.a].

6.2 Classification

Theorem 6.2. Let $N$$N$ be a closed oriented homologically $k$$k$-connected $n$$n$-manifold, $k\ge0$$k\ge0$. Then the Whitney invariant

$\displaystyle W:E^{2n-k}(N)\to H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$ $$\displaystyle W:E^{2n-k}(N)\to H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$$

is a bijection, provided $n\ge2k+4$$n\ge2k+4$ in the smooth category or $n\ge k+3$$n\ge k+3$ in the PL category.

This was proved for $k$$k$-connected manifolds in the smooth category [Haefliger&Hirsch1963, Theorem 2.4], and in the PL category in [Weber1967], [Hudson1969, $\S$$\S$11], cf. [Boechat&Haefliger1970, Theorem 1.6], [Boechat1971, Theorem 4.2], [Vrabec1977, Theorems 1.1 and 1.2], [Adachi1993, $\S$$\S$7]. The proof actually used the homological $k$$k$-connectedness assumption (basically because the $k$$k$-connectedness was used to ensure high enough connectedness of the complement in $\Rr^m$$\Rr^m$ to the image of $N$$N$, by Alexander duality and simple connectedness of the complement, so homological $k$$k$-connectedness of $N$$N$ is sufficient).

For $k=0$$k=0$ Theorem 6.2 is covered by Theorem 2.1; for $k\ge1$$k\ge1$ it is not. For $k+3\le n\le2k+1$$k+3\le n\le2k+1$ the PL case of Theorem 6.2 gives nothing but the Zeeman Unknotting Spheres Theorem [Skopenkov2016c, Theorem 2.3]. For the case of knotted tori see [Skopenkov2016k, Theorem 3.1].

An inverse to the map $W$$W$ of Theorem 6.2 is given by Example 6.1.

Because of the existence of knotted spheres the analogues of Theorem 6.2 for $n=k+2$$n=k+2$ in the PL case, and for $n\le2k+3$$n\le2k+3$ in the smooth case are false. So for the smooth category, $n\le2k+3$$n\le2k+3$ and $N$$N$ closed connected, a classification of $E^{2n-k}(N)$$E^{2n-k}(N)$ is much harder: for 40 years the only known complete readily calculable classification results were for homology spheres $N$$N$. E.g. $E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)}$$E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)}$ for any $s\ge2$$s\ge2$ [Haefliger1966, Corollary 8.14], [Skopenkov2016s, Theorem 3.2]. The following result for $n=2k+3$$n=2k+3$ was obtained using the Boéchat-Haefliger formula for the smoothing obstruction [Boechat&Haefliger1970, Theorem 2.1], [Boechat1971, Theorem 5.1]. Using that formula one can define the higher-dimensional Kreck invariant [Skopenkov2008, $\S$$\S$4].

Theorem 6.3 [Skopenkov2008, Higher-dimensional Classification Theorem]. Let $N$$N$ be a closed orientable homologically $(2l-2)$$(2l-2)$-connected $(4l-1)$$(4l-1)$-manifold. Then the Whitney invariant

$\displaystyle W:E^{6l}_D(N)\to H_{2l-1}(N)$ $$\displaystyle W:E^{6l}_D(N)\to H_{2l-1}(N)$$

is surjective and for any $u\in H_{2l-1}(N)$$u\in H_{2l-1}(N)$ the Kreck invariant

$\displaystyle \eta_u:W^{-1}u\to\Zz_{d(u)}$ $$\displaystyle \eta_u:W^{-1}u\to\Zz_{d(u)}$$

is a 1-1 correspondence, where $d(u)$$d(u)$ is the divisibility of the projection of $u$$u$ to the free part of $H_1(N)$$H_1(N)$.

Recall that the divisibility of zero is zero and the divisibility of $x\in G-\{0\}$$x\in G-\{0\}$ is $\max\{d\in\Zz\ | \ \text{there is }x_1\in G: \ x=dx_1\}$$\max\{d\in\Zz\ | \ \text{there is }x_1\in G: \ x=dx_1\}$.

How does one describe $E^m(N)$$E^m(N)$ when $N$$N$ is not $(2n-m)$$(2n-m)$-connected? For general $N$$N$ see the sentence on $E^{2n-1}(N)$$E^{2n-1}(N)$ at the end of $\S$$\S$2. We can say more as the connectivity $k$$k$ of $N$$N$ increases. Some estimations of $E^{2n-k-1}(N)$$E^{2n-k-1}(N)$ for a closed $k$$k$-connected $n$$n$-manifold $N$$N$ are presented in [Skopenkov2010]. For $k>1$$k>1$ one can go even further:

Theorem 6.4 [Becker&Glover1971, Corollary 1.3]. Let $N$$N$ be a closed $k$$k$-connected $n$$n$-manifold embeddable into $\Rr^m$$\Rr^m$, $m\ge2n-2k+1$$m\ge2n-2k+1$ and $2m\ge 3n+4$$2m\ge 3n+4$. Then there is a 1-1 correspondence

$\displaystyle E^m(N)\to [N_0, V_{m,n+1}].$ $$\displaystyle E^m(N)\to [N_0, V_{m,n+1}].$$

The 1-1 correspondence is constructed using `the Haefliger-Wu invariant' involving the configuration space of distinct pairs [Skopenkov2006, $\S$$\S$5]. For $k=0$$k=0$ Theorem 6.4 is the same as General Position Theorem [Skopenkov2016c, Theorem 2.1] (because $V_{2n+1,n+1}$$V_{2n+1,n+1}$ is $(n-1)$$(n-1)$-connected). For $k=1$$k=1$ Theorem 6.4 is covered by Theorem 6.2; for $k\ge2$$k\ge2$ it is not. For application to knotted tori see [Skopenkov2016k, Theorem 5.1]. For generalization to arbitrary manifolds see survey [Skopenkov2006, $\S$$\S$5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Observe that in Theorem 6.4 $V_{m,n+1}$$V_{m,n+1}$ can be replaced by $V_{M,M+n-m+1}$$V_{M,M+n-m+1}$ for any $M>n$$M>n$.

7 An orientation on the self-intersection set

Let $f:N\to\Rr^m$$f:N\to\Rr^m$ be a smooth map from an oriented $n$$n$-manifold $N$$N$ where $m\ge n+2$$m\ge n+2$. We assume that the closure

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${\rm Cl}\Sigma(f)$ of the self-intersection set of $f$$f$ has codimension 2 singularities, i.e., there is

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$P\subset{\rm Cl}\Sigma(f)$ such that

• both $P$$P$ and

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  ${\rm Cl}\Sigma(f)$ are subpolyhedra of some triangulation of $N$$N$,
• we have $\dim P\le\dim\Sigma(f)-2$$\dim P\le\dim\Sigma(f)-2$ and
• $\Sigma(f)-P$$\Sigma(f)-P$ is an open manifold consisting of self-transverse double points of $f$$f$.

Definition 7.1 (A canonical orientation on $\Sigma(f)-P$$\Sigma(f)-P$). Take points $x,y\in N$$x,y\in N$ away from $P$$P$ and such that $fx=fy$$fx=fy$. Then a $(2n-m)$$(2n-m)$-base $\xi_x$$\xi_x$ tangent to $\Sigma(f)-P$$\Sigma(f)-P$ at $x$$x$ gives a $(2n-m)$$(2n-m)$-base $\xi_y:=df_y^{-1}df_x(\xi_x)$$\xi_y:=df_y^{-1}df_x(\xi_x)$ tangent to $\Sigma(f)-P$$\Sigma(f)-P$ at $y$$y$. Since $N$$N$ is oriented, we can take positive $(m-n)$$(m-n)$-bases $\eta_x$$\eta_x$ and $\eta_y$$\eta_y$ at $x$$x$ and $y$$y$ normal to $\xi_x$$\xi_x$ and to $\xi_y$$\xi_y$. If the base $(df_x(\xi_x),df_x(\eta_x),df_y(\eta_y))$$(df_x(\xi_x),df_x(\eta_x),df_y(\eta_y))$ of $\Rr^m$$\Rr^m$ is positive, then call the base $\xi_x$$\xi_x$ positive. This is well-defined because a change of the sign of $\xi_x$$\xi_x$ forces changes of the signs of $\xi_y,\eta_x$$\xi_y,\eta_x$ and $\eta_y$$\eta_y$.

Remark 7.2 (Properties of the orientation just defined on $\Sigma(f) - P$$\Sigma(f) - P$)..

1. A change of the orientation of $N$$N$ forces changes of the signs of $\eta_x$$\eta_x$ and $\eta_y$$\eta_y$ and so does not change the orientation of $\Sigma(f)-P$$\Sigma(f)-P$.
2. The orientation on $\Sigma(f)-P$$\Sigma(f)-P$ need not extend to

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   ${\rm Cl}\Sigma(f)$: take the smooth cone $f:D^3\to\Rr^5$$f:D^3\to\Rr^5$ over a general position map $g:S^2\to\Rr^4$$g:S^2\to\Rr^4$ having only two transverse self-intersection points, where the smooth cone is defined by $f(tx):=(g(x)\sin(\pi t/2),\cos(\pi t/2))$$f(tx):=(g(x)\sin(\pi t/2),\cos(\pi t/2))$, for $x\in S^2$$x\in S^2$ and $t\in[0,1]$$t\in[0,1]$.
3. The orientation on $\Sigma(f)-P$$\Sigma(f)-P$ extends to

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   ${\rm Cl}\Sigma(f)$ if $m-n$$m-n$ is odd [Hudson1969, Lemma 11.4].

Remark 7.3 (A canonical orientation on $f\Sigma(f)-fP$$f\Sigma(f)-fP$ for $m-n$$m-n$ even). This remark is added as a complement for Definition 7.1 but is not needed for the definition of the Whitney invariant.

Take a $(2n-m)$$(2n-m)$-base $\xi$$\xi$ at a point $x\in f\Sigma(f)-fP$$x\in f\Sigma(f)-fP$. Since $N$$N$ is oriented, we can take a positive $(m-n)$$(m-n)$-base $\eta_+$$\eta_+$ normal to $f\Sigma(f)$$f\Sigma(f)$ in one sheet of $fN$$fN$. Analogously construct an $(m-n)$$(m-n)$-base $\eta_-$$\eta_-$ for the other sheet of $fN$$fN$. Since $m-n$$m-n$ is even, the orientation of the base $(\xi,\eta_+,\eta_-)$$(\xi,\eta_+,\eta_-)$ of $\Rr^m$$\Rr^m$ does not depend on choosing the first and the other sheet of $fN$$fN$ at $x$$x$. If the base $(\xi,\eta_+,\eta_-)$$(\xi,\eta_+,\eta_-)$ is positive, then call the base $\xi$$\xi$ positive. This is well-defined because a change of the sign of $\xi$$\xi$ forces changes of the signs of $\eta_+,\eta_-$$\eta_+,\eta_-$ and so of $(\xi,\eta_+,\eta_-)$$(\xi,\eta_+,\eta_-)$.

We remark that a change of the orientation of $N$$N$ forces changes of the signs of $\eta_+,\eta_-$$\eta_+,\eta_-$ and so does not change the orientation of $f\Sigma(f)-fP$$f\Sigma(f)-fP$.

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• [Weber1967] C. Weber, Plongements de polyedres dans le domaine metastable, Comment. Math. Helv. 42 (1967), 1-27. MR0238330 (38 #6606) Zbl 0152.22402
• [Yasui1984] T. Yasui, Enumerating embeddings of $n$$n$-manifolds in Euclidean $(2n-1)$$(2n-1)$-space, J. Math. Soc. Japan 36 (1984), no.4, 555–576. MR759414 Zbl 0557.57019

.5]{Avvakumov2017}. (Here 'linked' means that the images of the embeddings are not contained in disjoint cubes, unlike for the [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Embedded connected sum|unlinked embedded connected sum]] \cite[$\S]{Skopenkov2016c}.) For $a\in\Zz$ we repeat the above construction of $g_1$ replacing \partial D^{n+1}\times1_{n-1}$ by $|a|$ copies $(1+\frac1k)\partial D^{n+1}\times1_{n-1}$ of $S^n$, $k=1,\ldots,|a|$. The copies are outside $D^{n+1}\times S^{n-1}$ and are `parallel' to $\partial D^{n+1}\times1_{n-1}$. The copies have the standard orientation for $a>0$ or the opposite orientation for $a<0$. Then we make embedded connected sum along natural segments joining every $k$-th copy to the $(k+1)$-th copy. We obtain an embedding $g_a:S^n\to\Rr^{2n}$ which has disjoint images with $\Hud_n(0)$. Let $\Hud_n(a)$ be the linked embedded connected sum of $\Hud_n(0)$ and $g_a$. {{endthm}} The original motivation for Hudson was that $\Hud_n(1)$ is not isotopic to $\Hud_n(0)$ for any $n\ge3$ (this is a particular case of Proposition \ref{pr3} below). One might guess that $\Hud_n(a)$ is not isotopic to $\Hud_n(a')$ for $a\ne a'$ and that a $\Zz$-valued invariant of $E^{2n}(S^1 \times S^{n-1})$ can be defined by the homotopy class of the map $$S^n\overset g\to S^{2n}-{\rm i}(D^{n+1}\times S^{n-1})\sim S^{2n}-S^{n-1}\sim S^n \quad\text{which is}\quad a\in\pi_n(S^n)\cong\Zz.$$ However, this is only true for $n$ odd. {{beginthm|Proposition}}\label{pr3} For $n\ge3$ odd $\Hud_n(a)$ is isotopic to $\Hud_n(a')$ if and only if $a=a'$. For $n\ge4$ even $\Hud_n(a)$ is isotopic to $\Hud_n(a')$ if and only if $a\equiv a'\text{ mod}2$. {{endthm}} Proposition \ref{pr3} follows by calculation of [[#The_Whitney_invariant|the Whitney invariant]] (Remark \ref{re5}.d below) and, for $n$ even, by Theorem \ref{th4}. This proposition holds with the same proof in the piecewise smooth category, whose definition is recalled in \cite[Remark 1.1]{Skopenkov2016f}). Proposition \ref{pr3} also holds in the PL category (with an analogous construction of $\Hud_n(a)$ for the PL category). It would be interesting to find an explicit construction of an isotopy between $\Hud_{2k}(a)$ and $\Hud_{2k}(a+2)$, cf. \cite[$\S]{Vrabec1977}. Analogously, $\Hud_2(a)$ is not isotopic to $\Hud_2(a')$ if $a\not\equiv a'\text{ mod}2$. It would be interesting to know if the converse holds, e.g. is $\Hud_2(0)$ (PS or smoothly) isotopic to $\Hud_2(2)$. {{beginthm|Example}}\label{dh2n} Take any $a\in\Zz$. Take a map $\overline a:S^{n-1}\to S^{n-1}$ of degree $a$ (so we can take $\overline 1=\id$). Recall that $D^{n+1}=\{(y,x)\in D^n\times D^1\ :\ |y|^2+|x|^2\le1\}$. Define the smooth embedding $\Hud_n'(a)$ to be the composition $$S^1\times S^{n-1}\overset{\widehat a}\to D^{n+1}\times S^{n-1}\overset{\rm i}\to\Rr^{2n} \quad\text{where}\quad \widehat a(e^{i\theta},y):=(\overline a(y)\cos\theta,\sin\theta,y).$$ [[Image:hudson_torus_new.jpg|thumb|450px|Figure 1: The Hudson torus]] Let us present a geometric description of this embedding. Define a map $\widetilde a:S^0\times S^{n-1}\to S^{n-1}$ by $\widetilde a(s,t):=s\overline a(t)$.

\in D^n$ on one component N is a connected manifold of dimension $n>1$$n>1$, and $m \ge2n+1$$m \ge2n+1$, then every two embeddings $N \to\Rr^m$$N \to\Rr^m$ are isotopic [Skopenkov2016c, Theorem 3.2], [Skopenkov2006, Theorem 2.5]. In this page we summarize the situation for $m=2n\ge6$$m=2n\ge6$ and $N$$N$ is a connected, as well as in some more general situations. For the classification of embeddings of some disconnected manifolds see [Skopenkov2016h].

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$$\S$1, $\S$$\S$3]. Denote $1_n:=(1,0,\ldots,0)\in S^n$$1_n:=(1,0,\ldots,0)\in S^n$.

2 Classification

For the next theorem, the Whitney invariant $W$$W$ is defined in $\S$$\S$5 below.

Theorem 2.1. Assume that $N$$N$ is a closed connected $n$$n$-manifold, and either $n\ge4$$n\ge4$ or $n=3$$n=3$ and we are in the PL category.

(a) If $N$$N$ is oriented, the Whitney invariant,

$\displaystyle W:E^{2n}(N)\to H_1(N;\Zz_{\varepsilon(n-1)}),$ $$\displaystyle W:E^{2n}(N)\to H_1(N;\Zz_{\varepsilon(n-1)}),$$

is a 1-1 correspondence.

(b) If $N$$N$ is non-orientable, then there is a 1-1 correspondence

$\displaystyle E^{2n}(N)\to \begin{cases} H_1(N;\Zz_2) & n\text{ is odd}\\ \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s & n\text{ is even}\end{cases}.$ $$\displaystyle E^{2n}(N)\to \begin{cases} H_1(N;\Zz_2) & n\text{ is odd}\\ \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s & n\text{ is even}\end{cases}.$$

Remark 2.2 (Comments on the proof). Part (a) is proved in [Haefliger&Hirsch1963, Theorem 2.4] in the smooth category, and in [Weber1967, Theorem 4' in $\S$$\S$2], [Hudson1969, $\S$$\S$11], [Vrabec1977, Theorems 1.1 and 1.2] in the PL category, see also [Haefliger1962b, 1.3.e], [Haefliger1963], [Bausum1975, Theorem 43].

Part (b) is proved in [Bausum1975, Theorem 43] in the smooth category. By [Weber1967, Theorems 1 and 1' in $\S$$\S$2], [Skopenkov1997, Theorem 1.1.c] the proof works also in the PL category.

In Part (b) we replaced the kernel $\ker Sq^1$$\ker Sq^1$ from [Bausum1975, Theorem 43] by $\Zz_2^{s-1}$$\Zz_2^{s-1}$. This is possible because, as a specialist could see, $Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$$Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$ is given by multiplication with the first Stefel-Whitney class $w_1(N)$$w_1(N)$ (which equals to the first Wu class $v_1(N)$$v_1(N)$ [Milnor&Stasheff1974, Theorem 11.4]). Since $N$$N$ is non-orientable, $w_1(N)\neq 0$$w_1(N)\neq 0$. So by Poincaré duality, $\ker Sq^1 \cong \Zz_2^{s-1}$$\ker Sq^1 \cong \Zz_2^{s-1}$.

The 1-1 correspondence from (b) can presumably be defined as a (generalized) Whitney invariant, see [Vrabec1977], but the proof used 'the Haefliger-Wu invariant' whose definition can be found e.g. in [Skopenkov2006, $\S$$\S$5]. It would be interesting to check if part (b) is equivalent to different forms of description of $E^{2n}(N)$$E^{2n}(N)$ [Haefliger1962b, 1.3.e], [Haefliger1963], [Vrabec1977, Theorem 1.1].

The classification of smooth embeddings of 3-manifolds in $\Rr^6$$\Rr^6$ is more complicated, see Theorem 6.3 below for $l=1$$l=1$ or [Skopenkov2016t].

Concerning embeddings of connected $n$$n$-manifolds in $\Rr^{2n-1}$$\Rr^{2n-1}$ see [Yasui1984] for $n\ge5$$n\ge5$, [Skopenkov2016f] for $n=4$$n=4$, and [Saeki1999], [Skopenkov2010], [Tonkonog2010] for manifolds with boundary.

Theorem 2.1 is generalized to a description of $E^{2n-k}(N)$$E^{2n-k}(N)$ for closed $k$$k$-connected $n$$n$-manifolds $N$$N$, see Theorem 6.2.

3 Hudson and Lagrangian tori

Together with the Haefliger knotted sphere [Skopenkov2016t, Example 2.1], [Skopenkov2006, Example 3.4], the examples of Hudson tori presented below were the first examples of non-isotopic embeddings in codimension greater than 2. (Hudson's construction [Hudson1963] was not as explicit as those below.) Abbreviate

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${\rm i}_{2n,n-1}$ to just

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${\rm i}$.

Example 3.1. Let us construct, for any $a\in\Zz$$a\in\Zz$ and $n\ge2$$n\ge2$, a smooth embedding

$\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.$ $$\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.$$

We start with the cases $a=0,1$$a=0,1$.

Take the standard inclusion $\partial D^2\subset\partial D^{n+1}$$\partial D^2\subset\partial D^{n+1}$. The 'standard embedding' $\Hud_n(0)$$\Hud_n(0)$ is given by the standard inclusions

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$$\displaystyle \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\overset{{\rm i}}\to\Rr^{2n}.$$

Define the `standard embedding'

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$2{\rm i}:2D^{n+1}\times S^{n-1}\to\Rr^{2n}$ analogously to

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${\rm i}$, where $2$$2$ means homothety with coefficient 2.

Take the embedding $g_1$$g_1$ given by

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$$\displaystyle S^n=2\partial D^{n+1}\times1_{n-1}\subset 2D^{n+1}\times S^{n-1}\overset{2{\rm i}}\to\Rr^{2n}.$$

The segment

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$2{\rm i}([1,2]1_n\times1_{n-1})$ joins the images of $\Hud_n(0)$$\Hud_n(0)$ and $g_1$$g_1$; the interior of this segment misses the images. Let $\Hud_n(1)$$\Hud_n(1)$ be the linked embedded connected sum of $\Hud_n(0)$$\Hud_n(0)$ and $g_1$$g_1$ along this segment, compatible with the orientation, cf. [Avvakumov2017, $\S$$\S$1.5]. (Here 'linked' means that the images of the embeddings are not contained in disjoint cubes, unlike for the unlinked embedded connected sum [Skopenkov2016c, $\S$$\S$5].)

For $a\in\Zz$$a\in\Zz$ we repeat the above construction of $g_1$$g_1$ replacing $2\partial D^{n+1}\times1_{n-1}$$2\partial D^{n+1}\times1_{n-1}$ by $|a|$$|a|$ copies $(1+\frac1k)\partial D^{n+1}\times1_{n-1}$$(1+\frac1k)\partial D^{n+1}\times1_{n-1}$ of $S^n$$S^n$, $k=1,\ldots,|a|$$k=1,\ldots,|a|$. The copies are outside $D^{n+1}\times S^{n-1}$$D^{n+1}\times S^{n-1}$ and are `parallel' to $\partial D^{n+1}\times1_{n-1}$$\partial D^{n+1}\times1_{n-1}$. The copies have the standard orientation for $a>0$$a>0$ or the opposite orientation for $a<0$$a<0$. Then we make embedded connected sum along natural segments joining every $k$$k$-th copy to the $(k+1)$$(k+1)$-th copy. We obtain an embedding $g_a:S^n\to\Rr^{2n}$$g_a:S^n\to\Rr^{2n}$ which has disjoint images with $\Hud_n(0)$$\Hud_n(0)$. Let $\Hud_n(a)$$\Hud_n(a)$ be the linked embedded connected sum of $\Hud_n(0)$$\Hud_n(0)$ and $g_a$$g_a$.

The original motivation for Hudson was that $\Hud_n(1)$$\Hud_n(1)$ is not isotopic to $\Hud_n(0)$$\Hud_n(0)$ for any $n\ge3$$n\ge3$ (this is a particular case of Proposition 3.2 below). One might guess that $\Hud_n(a)$$\Hud_n(a)$ is not isotopic to $\Hud_n(a')$$\Hud_n(a')$ for $a\ne a'$$a\ne a'$ and that a $\Zz$$\Zz$-valued invariant of $E^{2n}(S^1 \times S^{n-1})$$E^{2n}(S^1 \times S^{n-1})$ can be defined by the homotopy class of the map

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$$\displaystyle S^n\overset g\to S^{2n}-{\rm i}(D^{n+1}\times S^{n-1})\sim S^{2n}-S^{n-1}\sim S^n \quad\text{which is}\quad a\in\pi_n(S^n)\cong\Zz.$$

However, this is only true for $n$$n$ odd.

Proposition 3.2. For $n\ge3$$n\ge3$ odd $\Hud_n(a)$$\Hud_n(a)$ is isotopic to $\Hud_n(a')$$\Hud_n(a')$ if and only if $a=a'$$a=a'$.

For $n\ge4$$n\ge4$ even $\Hud_n(a)$$\Hud_n(a)$ is isotopic to $\Hud_n(a')$$\Hud_n(a')$ if and only if $a\equiv a'\text{ mod}2$$a\equiv a'\text{ mod}2$.

Proposition 3.2 follows by calculation of the Whitney invariant (Remark 5.3.d below) and, for $n$$n$ even, by Theorem 2.1. This proposition holds with the same proof in the piecewise smooth category, whose definition is recalled in [Skopenkov2016f, Remark 1.1]). Proposition 3.2 also holds in the PL category (with an analogous construction of $\Hud_n(a)$$\Hud_n(a)$ for the PL category). It would be interesting to find an explicit construction of an isotopy between $\Hud_{2k}(a)$$\Hud_{2k}(a)$ and $\Hud_{2k}(a+2)$$\Hud_{2k}(a+2)$, cf. [Vrabec1977, $\S$$\S$5]. Analogously, $\Hud_2(a)$$\Hud_2(a)$ is not isotopic to $\Hud_2(a')$$\Hud_2(a')$ if $a\not\equiv a'\text{ mod}2$$a\not\equiv a'\text{ mod}2$. It would be interesting to know if the converse holds, e.g. is $\Hud_2(0)$$\Hud_2(0)$ (PS or smoothly) isotopic to $\Hud_2(2)$$\Hud_2(2)$.

Example 3.3. Take any $a\in\Zz$$a\in\Zz$. Take a map $\overline a:S^{n-1}\to S^{n-1}$$\overline a:S^{n-1}\to S^{n-1}$ of degree $a$$a$ (so we can take $\overline 1=\id$$\overline 1=\id$). Recall that $D^{n+1}=\{(y,x)\in D^n\times D^1\ :\ |y|^2+|x|^2\le1\}$$D^{n+1}=\{(y,x)\in D^n\times D^1\ :\ |y|^2+|x|^2\le1\}$. Define the smooth embedding $\Hud_n'(a)$$\Hud_n'(a)$ to be the composition

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$$\displaystyle S^1\times S^{n-1}\overset{\widehat a}\to D^{n+1}\times S^{n-1}\overset{\rm i}\to\Rr^{2n} \quad\text{where}\quad \widehat a(e^{i\theta},y):=(\overline a(y)\cos\theta,\sin\theta,y).$$

Figure 1: The Hudson torus

Let us present a geometric description of this embedding. Define a map $\widetilde a:S^0\times S^{n-1}\to S^{n-1}$$\widetilde a:S^0\times S^{n-1}\to S^{n-1}$ by $\widetilde a(s,t):=s\overline a(t)$$\widetilde a(s,t):=s\overline a(t)$. This map gives an embedding

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$$\displaystyle S^0\times S^{n-1}\xrightarrow{\widetilde a\times {\rm pr}_2}S^{n-1}\times S^{n-1}\subset D^{n+1}\times S^{n-1}\overset{{\rm i}}\to\Rr^{2n}.$$

See Figure 1. The image of

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$\widetilde a\times{\rm pr}_2$ is the union of the graphs of the maps $\overline a$$\overline a$ and $-\overline a$$-\overline a$. For any $t\in S^{n-1}$$t\in S^{n-1}$ the disk

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${\rm i}(D^{n+1}\times t)$ intersects the image at two points lying in

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${\rm i}(D^n\times t)$, i.e., at the image of an embedding

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$S^0\times t\to {\rm i}(D^n\times t)$. The embedding $\Hud_n'(a)$$\Hud_n'(a)$ is obtained by extending the latter embeddings to embeddings

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$S^1\times t\to {\rm i}(D^{n+1}\times t)$ for all $t$$t$. See Figure 2.

Figure 2: To the construction of the Hudson torus

Remark 3.4. (a) The analogue of Proposition 3.2 for $\Hud_n$$\Hud_n$ replaced to $\Hud_n'$$\Hud_n'$ holds, with an analogous proof.

(b) The embeddings $\Hud_n(a)$$\Hud_n(a)$ and $\Hud_n'(a)$$\Hud_n'(a)$ are smoothly isotopic for $n\ge4$$n\ge4$ and are PS isotopic for $n\ge3$$n\ge3$ [Skopenkov2006a, commutativity of the left upper square in the Restriction Lemma 5.2], [Skopenkov2015a, Lemma 2.15.c] (see [Skopenkov2016f, Remark 1.2]). This follows by calculation of the Whitney invariant (Remark 5.3.d below). It would be interesting to know if they are smoothly isotopic for $n=3$$n=3$. It would be interesting to know if they are piecewise smoothly isotopic for $n=2$$n=2$.

(b') Although construction of $\Hud_n(a)$$\Hud_n(a)$ and $\Hud_n'(a)$$\Hud_n'(a)$ is explicit, no explicit construction of an isotopy between them is known. Proof from (b) of their being isotopic is by calculating invariants, and using completeness result for the invariants. This proof is fairly constructive, so in principle it does give a specific isotopy. However, the construction is so complicated that one would not call such an isotopy explicit. This is analogous to the Smale `turning sphere inside out' theorem (although the Smale theorem concerns regular homotopy of immersions not isotopy of embeddings).

In higher dimensions, a proof that any embedding is isotopic to some of the constructed embeddings is even less likely to use an explicit construction. The only proofs we have go through calculating the invariants and using classification results.

(c) For $n=2$$n=2$ Example 3.3 gives what we call the 'left' Hudson torus. The 'right' Hudson torus is constructed analogously. It is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of $S^1\times S^1$$S^1\times S^1$. The right and the left Hudson tori are not isotopic by Remark 5.3.d below.

(d) Analogously one constructs the Hudson torus $\Hud_{n,p}(a):S^p\times S^{n-p}\to\Rr^{2n-p+1}$$\Hud_{n,p}(a):S^p\times S^{n-p}\to\Rr^{2n-p+1}$ for $a\in\Zz$$a\in\Zz$ and $n>p\ge0$$n>p\ge0$ or, more generally, $\Hud_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$$\Hud_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$ for $a\in\pi_n(S^{m-n+p-1})$$a\in\pi_n(S^{m-n+p-1})$ and $m>n>p\ge0$$m>n>p\ge0$. There are versions $\Hud'_{m,n,p}(a)$$\Hud'_{m,n,p}(a)$ of these constructions corresponding to Definition 3.3. For $p=0$$p=0$ this corresponds to the Zeeman map [Skopenkov2016h, Definition 2.2] and its composition with 'the unframed second Kirby move' [Skopenkov2015a, $\S$$\S$2.3]. It would be interesting to know if the links $\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$$\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$ are isotopic, cf. [Skopenkov2015a, Remark 2.7.b]. These constructions can be further generalized [Skopenkov2016k].

Example 3.5. Define a smooth embedding

$\displaystyle l:S^1\times S^{n-1}\to \C^n=\R^{2n}\quad\text{by}\quad l(z,x) := \left(1+\frac z2\right)x.$ $$\displaystyle l:S^1\times S^{n-1}\to \C^n=\R^{2n}\quad\text{by}\quad l(z,x) := \left(1+\frac z2\right)x.$$

Here $S^1\subset\R^2=\C$$S^1\subset\R^2=\C$ and $S^{n-1}\subset\R^n\subset\C^n$$S^{n-1}\subset\R^n\subset\C^n$. This embedding is known to be Lagrangian; see more in [Nemirovski2024].

4 Action by linked embedded connected sum

In this section we generalize the construction of the Hudson torus $\Hud(a)$$\Hud(a)$. Let $N$$N$ be a closed connected oriented $n$$n$-manifold. We work in the smooth category which we omit. Apparently analogous results hold for $n\ge3$$n\ge3$ in the PL and PS categories (see [Skopenkov2016f, Remark 1.2]).

Example 4.1. For any $n\ge4$$n\ge4$, an embedding $f_0:N\to\Rr^{2n}$$f_0:N\to\Rr^{2n}$ and $a\in H_1(N;\Zz)$$a\in H_1(N;\Zz)$, we shall construct an embedding $f_a:N\to\Rr^{2n}$$f_a:N\to\Rr^{2n}$. This embedding is said to be obtained by linked embedded connected sum of $f_0$$f_0$ with an $n$$n$-sphere representing the `homology Alexander dual' $A:=\widehat{A_{f_0}}a\in H_n(C_{f_0})$$A:=\widehat{A_{f_0}}a\in H_n(C_{f_0})$ of $a$$a$ (defined in [Skopenkov2005, Alexander Duality Lemma 4.6]).

Represent $a$$a$ by an embedding $a:S^1\to N$$a:S^1\to N$. By definition, the class $A$$A$ is represented by properly oriented $\nu_{f_0}^{-1}a(S^1)$$\nu_{f_0}^{-1}a(S^1)$. Since any orientable bundle over $S^1$$S^1$ is trivial, $\nu_{f_0}^{-1}a(S^1)\cong S^1\times S^{n-1}$$\nu_{f_0}^{-1}a(S^1)\cong S^1\times S^{n-1}$. Take an embedding $g:S^1\times S^{n-1}\to C_{f_0}$$g:S^1\times S^{n-1}\to C_{f_0}$ whose image is $\nu_{f_0}^{-1}a(S^1)$$\nu_{f_0}^{-1}a(S^1)$ and which represents $A$$A$. By embedded surgery on $S^1\times1_{n-1}\subset S^1\times S^{n-1}$$S^1\times1_{n-1}\subset S^1\times S^{n-1}$ we obtain an embedding $g_1:S^n\to C_{f_0}$$g_1:S^n\to C_{f_0}$ representing $A$$A$ (see details in Proposition 4.2 below). Define $f_a$$f_a$ to be the linked embedded connected sum of $f_0$$f_0$ and $g_1$$g_1$, along some arc joining their images.

Proposition 4.2 (Embedded surgery). For any $n\ge3$$n\ge3$, a neighborhood $U$$U$ of a codimension at least 3 subpolyhedron in $\Rr^{2n}$$\Rr^{2n}$ and an embedding $g:S^1\times S^{n-1}\to\Rr^{2n}-U$$g:S^1\times S^{n-1}\to\Rr^{2n}-U$ there is an embedding $g_1:S^n\to\Rr^{2n}-U$$g_1:S^n\to\Rr^{2n}-U$ homologous to $g$$g$.

Proof. Take a vector field on $g(S^1\times1_{n-1})$$g(S^1\times1_{n-1})$ normal to $g(S^1\times S^{n-1})$$g(S^1\times S^{n-1})$. Extend $g|_{S^1\times1_{n-1}}$$g|_{S^1\times1_{n-1}}$ along this vector field to a map $\overline b:D^2\to\Rr^{2n}$$\overline b:D^2\to\Rr^{2n}$.

Since $2n>4$$2n>4$ and $U$$U$ is a neighborhood $U$$U$ of a codimension at least 3 subpolyhedron, by general position we may assume that $\overline b$$\overline b$ is an embedding and that

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$\overline b({\rm Int}D^2)$ misses $U\cup g(S^1\times S^{n-1})$$U\cup g(S^1\times S^{n-1})$.

Since $n-1>1$$n-1>1$, we have $\pi_1(V_{2n-2,n-1})=0$$\pi_1(V_{2n-2,n-1})=0$. Hence the standard $(n-1)$$(n-1)$-framing of $S^1\times1_{n-1}$$S^1\times1_{n-1}$ in $S^1\times S^{n-1}$$S^1\times S^{n-1}$ extends to an $(n-1)$$(n-1)$-framing on $\overline b(D^2)$$\overline b(D^2)$ in $\Rr^{2n}$$\Rr^{2n}$. Thus $\overline b$$\overline b$ extends to an embedding

$\displaystyle \widehat b:D^2\times D^{n-1}\to\R^{2n}-U\quad\text{such that}\quad \widehat b(\partial D^2\times D^{n-1})\subset g(S^1\times S^{n-1}).$ $$\displaystyle \widehat b:D^2\times D^{n-1}\to\R^{2n}-U\quad\text{such that}\quad \widehat b(\partial D^2\times D^{n-1})\subset g(S^1\times S^{n-1}).$$

Take an embedding $g_1:S^n\to\R^{2n}-U$$g_1:S^n\to\R^{2n}-U$ such that

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$$\displaystyle g_1(S^n)\ =\ g(S^1\times S^{n-1})-\widehat b(\partial D^2\times{\rm Int}D^{n-1}) \bigcup\limits_{\widehat b(\partial D^2\times\partial D^{n-1})} \widehat b(D^2\times\partial D^{n-1})\ \cong\ S^n.$$

with proper orientation so that $g_1$$g_1$ is homologous to $g$$g$. QED

The isotopy class of the embedding $f_a$$f_a$ is independent of the choises in the construction. The independence of the arc and of the maps $g,g_1$$g,g_1$ follows by $n\ge3$$n\ge3$ and by Proposition 4.3 below, respectively.

By Definition 5.1 of the Whitney invariant, $W(f_a,f_0)$$W(f_a,f_0)$ is $a$$a$ for $n\ge3$$n\ge3$ odd and $a \mod2$$a \mod2$ for $n\ge4$$n\ge4$ even. Thus by Theorem 2.1.a for $n\ge4$$n\ge4$ all isotopy classes of embeddings $N\to\Rr^{2n}$$N\to\Rr^{2n}$ can be obtained from any chosen embedding $f_0$$f_0$ by the above construction.

Proposition 4.3. For any $n\ge4$$n\ge4$ both the linked embedded connected sum and parametric connected sum (introduced in [Skopenkov2006a], [Skopenkov2015a]) define free transitive actions of $H_1(N;\Zz_{\varepsilon(n-1)})$$H_1(N;\Zz_{\varepsilon(n-1)})$ on $E^{2n}(N)$$E^{2n}(N)$.

This follows by Theorem 2.1.a and by [Skopenkov2014, Remark 18.a].

5 The Whitney invariant

Let $N$$N$ be a closed $n$$n$-manifold. Take an embedding $f_0:N\to\Rr^m$$f_0:N\to\Rr^m$. Fix an orientation on $\Rr^m$$\Rr^m$. For any other embedding $f \colon N \to \Rr^m$$f \colon N \to \Rr^m$ we define the Whitney invariant

$\displaystyle W(f, f_0)=W_{f_0}(f)=W(f)\in H_{2n-m+1}(N;\Zz_N).$ $$\displaystyle W(f, f_0)=W_{f_0}(f)=W(f)\in H_{2n-m+1}(N;\Zz_N).$$

Here the coefficients $\Zz_N$$\Zz_N$ are $\Zz$$\Zz$ if $N$$N$ is oriented and $m-n$$m-n$ is odd, and are $\Zz_2$$\Zz_2$ otherwise.

Roughly speaking,

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$W(f):=[{\rm Cl}\Sigma(H)]$ is defined as the homology class of the closure of the self-intersection set of a general position homotopy $H$$H$ between $f$$f$ and $f_0$$f_0$.

This is formalized in Definition 5.2 in the smooth category, following [Skopenkov2010], see also [Haefliger&Hirsch1963]. The definition in the PL category is analogous [Hudson1969, $\S$$\S$11], [Vrabec1977, p. 145], [Skopenkov2006, $\S$$\S$2.4 `The Whitney invariant']. We begin by presenting a simpler definition, Definition 5.1, for a particular case.

For Theorem 2.1 only the case $m=2n$$m=2n$ is required.

Definition 5.1. Assume that $N$$N$ is $(2n-m)$$(2n-m)$-connected and $2m\ge3n+3$$2m\ge3n+3$. Then by [Haefliger&Hirsch1963, Theorem 3.1.b] restrictions of $f$$f$ and $f_0$$f_0$ to $N_0$$N_0$ are isotopic, cf. [Takase2006, Lemma 2.2]. (Here is sketch of an argument. Using the Smale-Hirsch classification of immersions we obtain that restrictions of $f$$f$ and $f_0$$f_0$ to $N_0$$N_0$ are `regular homotopic', see [Koschorke2013, Definition 2.7]. Since $N$$N$ is $(2n-m)$$(2n-m)$-connected, $N_0$$N_0$ retracts to an $(m-n-1)$$(m-n-1)$-dimensional polyhedron. Therefore these restrictions are isotopic.)

So we can make an isotopy of $f$$f$ and assume that $f=f_0$$f=f_0$ on $N_0$$N_0$. Take a general position homotopy $F:B^n\times I\to\Rr^m$$F:B^n\times I\to\Rr^m$ relative to $\partial B^n$$\partial B^n$ between the restrictions of $f$$f$ and $f_0$$f_0$ to $B^n$$B^n$. Let $f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$$f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$ (`the intersection of this homotopy with $f(N-B^n)$$f(N-B^n)$').

Since $n+2(n+1)<2m$$n+2(n+1)<2m$, by general position

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${\rm Cl}(f\cap F)$ is a compact $(2n+1-m)$$(2n+1-m)$-manifold whose boundary is contained in $\partial N_0$$\partial N_0$.

So $f\cap F$$f\cap F$ carries a homology class with $\Zz_2$$\Zz_2$ coefficients. If $m-n$$m-n$ is odd and $N$$N$ is oriented, then $f\cap F$$f\cap F$ has a natural orientation defined below, and so carries a homology class with $\Zz$$\Zz$ coefficients. Define $W(f)$$W(f)$ to be the homology class:

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$$\displaystyle W(f):=[{\rm Cl}(f\cap F)]\in H_{2n-m+1}(N_0,\partial N_0;\Zz_N)\cong H_{2n-m+1}(N;\Zz_N).$$

The orientation on $f\cap F$$f\cap F$ (extendable to

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${\rm Cl}(f\cap F)$) is defined (for $m-n$$m-n$ odd and $N$$N$ is oriented) as follows (cf. Remark 7.3). For any point $x_f\in f\cap F$$x_f\in f\cap F$ take a base $\xi_f$$\xi_f$ at $x_f$$x_f$ tangent to $f\cap F$$f\cap F$. Complete this base to a positive base $(\xi_f,\eta_f)$$(\xi_f,\eta_f)$ tangent to $N$$N$. Since $n+2(n+1)<2m$$n+2(n+1)<2m$, by general position there is a unique point $x_F\in B^n\times I$$x_F\in B^n\times I$ such that $Fx_F=fx_f$$Fx_F=fx_f$. The tangent base $\xi_f$$\xi_f$ at $x_f$$x_f$ thus gives a base $\xi_F$$\xi_F$ at $x_F$$x_F$ tangent to $B^n\times I$$B^n\times I$ such that $df(x_f)\xi_f=dF(x_F)\xi_F$$df(x_f)\xi_f=dF(x_F)\xi_F$. Complete this base $\xi_F$$\xi_F$ to a positive base $(\xi_F,\eta_F)$$(\xi_F,\eta_F)$ tangent to $B^n\times I$$B^n\times I$, where the orientation on $B^n$$B^n$ comes from $N$$N$. The union $\zeta=(df(x_f)\xi_f,df(x_f)\eta_f,dF(x_F)\eta_F)$$\zeta=(df(x_f)\xi_f,df(x_f)\eta_f,dF(x_F)\eta_F)$ of the images of the constructed bases is a base at $fx_f=Fx_F$$fx_f=Fx_F$ of $\Rr^m$$\Rr^m$. If $\zeta$$\zeta$ is positive, then call the tangent base $\xi_f$$\xi_f$ of $f\cap F$$f\cap F$ `positive'. Since a change of the orientation on $f\cap F$$f\cap F$ forces a change of the orientation of $\zeta$$\zeta$, this condition indeed defines an orientation on $f\cap F$$f\cap F$.

Definition 5.2. Assume that $m\ge n+2$$m\ge n+2$. Take a general position homotopy $H:N\times I\to\Rr^m\times I$$H:N\times I\to\Rr^m\times I$ between $f_0$$f_0$ and $f$$f$.

The closure

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${\rm Cl}\Sigma(H)$ of the self-intersection set carries a cycle mod 2. If $N$$N$ is oriented and $m-n$$m-n$ is odd, the closure also carries an integer cycle. See [Hudson1967, $\S$$\S$11], [Skopenkov2006, $\S$$\S$2.3 `The Whitney obstruction']. Let us informally explain these facts. For $2m\ge3n+2$$2m\ge3n+2$ by general position the closure

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${\rm Cl}\Sigma(H)$ can be assumed to be a submanifold. In general, since $m\ge n+2$$m\ge n+2$, by general position the closure has codimension 2 singularities, as defined in $\S$$\S$7. So the closure carries a cycle mod 2. When $m-n$$m-n$ is odd the closure also has a canonical orientation (see Definition 7.1 and Remark 7.2), so the closure carries an integer cycle.

Define the Whitney invariant to be the homology class:

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$$\displaystyle W(f):=[{\rm Cl}\Sigma(H)]\in H_{2n-m+1}(N\times I;\Zz_N)\cong H_{2n-m+1}(N;\Zz_N).$$

Clearly, $W(f) = W(f')$$W(f) = W(f')$ if $f$$f$ is isotopic to $f'$$f'$. Hence the Whitney invariant defines a map

$\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_N),\quad [f] \mapsto W(f).$ $$\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_N),\quad [f] \mapsto W(f).$$

Clearly, $W(f_0)=0$$W(f_0)=0$ (for both definitions).

The definition of $W$$W$ depends on the choice of $f_0$$f_0$, but we write $W$$W$ not $W_{f_0}$$W_{f_0}$ for brevity.

Remark 5.3. (a) The Whitney invariant is well-defined by Definition 5.2, i.e. is independent of the choice of a general position homotopy $H:N\times I\to\Rr^m\times I$$H:N\times I\to\Rr^m\times I$ from $f_0$$f_0$ to $f$$f$.

This follows from the equality

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$[{\rm Cl}\Sigma(H_0)]−[{\rm Cl}\Sigma(H_1)] = \partial [{\rm Cl}\Sigma(H_{01})]$ for a general position homotopy $H_{01}:N\times I\times I\to\Rr^m\times I\times I$$H_{01}:N\times I\times I\to\Rr^m\times I\times I$ between general position homotopies $H_0,H_1:N\times I\to\Rr^m\times I$$H_0,H_1:N\times I\to\Rr^m\times I$ from $f_0$$f_0$ to $f$$f$. See details in [Hudson1969, $\S$$\S$11].

(b) Definition 5.1 is a particular case of Definition 5.2. (Indeed, if $f=f_0$$f=f_0$ on $N_0$$N_0$, we can take $H$$H$ to be fixed on $N_0$$N_0$. See details in [Skopenkov2010, Difference Lemma 2.4].) Hence the Whitney invariant is well-defined by Definition 5.1, i.e. independent of the choice of $F$$F$ and of the isotopy making $f=f_0$$f=f_0$ outside $B^n$$B^n$.

(c) The class $W(f)$$W(f)$ is independent of the choice of the orientation on $N$$N$ (because a change of the orientation on $N$$N$ forces a change of the orientation on $f\cap F$$f\cap F$ or on

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${\rm Cl}\Sigma(H)$). For the reflection $\sigma:\Rr^m\to\Rr^m$$\sigma:\Rr^m\to\Rr^m$ with respect to a hyperplane we have $W(\sigma\circ f)=-W(f)$$W(\sigma\circ f)=-W(f)$ (because a change of the orientation on $\Rr^m$$\Rr^m$ forces a change of the orientation on $f\cap F$$f\cap F$ or on

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${\rm Cl}\Sigma(H)$; for Definition 5.1 also observe that we may assume that $f=f_0=\sigma\circ f$$f=f_0=\sigma\circ f$ on $N_0$$N_0$).

(d) For the Hudson tori $W(\Hud_n(a))=W(\Hud'_n(a))$$W(\Hud_n(a))=W(\Hud'_n(a))$ is $a$$a$ or $a\mod2$$a\mod2$ for $n\ge3$$n\ge3$, and $W(\Hud'_2(a))=(a\mod2,0)$$W(\Hud'_2(a))=(a\mod2,0)$.

For $\Hud_n(a)$$\Hud_n(a)$ this is clear by Definition 5.1. For $\Hud'_n(a)$$\Hud'_n(a)$ and $n\ge3$$n\ge3$ this was proved in [Hudson1963] (with a different but equivalent definition of the Whitney invariant; using and proving a particular case of Remark 5.3.f). For $\Hud'_2(a)$$\Hud'_2(a)$ the proof is analogous.

(e) $W(f\#g)=W(f)$$W(f\#g)=W(f)$ for any pair of embeddings $f:N\to\Rr^m$$f:N\to\Rr^m$ and $g:S^n\to\Rr^m$$g:S^n\to\Rr^m$. This is clear by Definition 5.1 because $W(f\#g)-W(f)=W(f\#g,f_0)-W(f,f_0)=W(f\#g,f)=0$$W(f\#g)-W(f)=W(f\#g,f_0)-W(f,f_0)=W(f\#g,f)=0$. Let us prove the latter equality. Take the identical isotopy $H_f$$H_f$ of $f$$f$ and a general position homotopy $H_g$$H_g$ between $g$$g$ and the standard embedding. Then the boundary connected sum $H_f\sharp H_g$$H_f\sharp H_g$ is a general position homotopy between $f\#g$$f\#g$ and an embedding isotopic to $f$$f$. The cycle

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${\rm Cl}\Sigma(H_f\sharp H_g)={\rm Cl}\Sigma H_g$ is null-homologous in $S^n$$S^n$ and hence in $N\cong N\#S^n$$N\cong N\#S^n$; cf. [Skopenkov2008, Addendum to the Classification Theorem].

(f) For $m=2n+1$$m=2n+1$ and $N=S^n\sqcup S^n$$N=S^n\sqcup S^n$ the Whitney invariant equals to the pair of linking coefficients [Skopenkov2016h, $\S$$\S$3].

(g) The Whitney invariant need not be a bijection for $m<2n$$m<2n$. This is seen, for example, by applying Theorem 6.4 below in case of knotted tori [Skopenkov2016k, Theorem 5.1]) or by taking $n$$n$ even, $N$$N$ non-orientable, $m=2n$$m=2n$ and applying by Theorem 2.1.b.

(h) The paper [Haefliger&Hirsch1963, $\S$$\S$2] gives yet another equivalent definition of the Whitney invariant, and proves its injectivity under certain restrictions (the Haefliger-Hirsch $h$$h$-principle, see Theorem 6.2).

(i) Let $N$$N$ be a closed orientable $n$$n$-manifold. For two embeddings $f,g:N\to\R^{2n-1}$$f,g:N\to\R^{2n-1}$ the invariant defined in [Haefliger&Hirsch1963, $\S$$\S$2] equals plus or minus their Whitney invariant $W(f,g)$$W(f,g)$ from Definition 5.1. This is proved analogously to the following version.

Let $D^n$$D^n$ and $B^n$$B^n$ be closed $n$$n$-balls in $N$$N$ such that

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$D^n\subset{\rm Int}B^n$. Let $f:N\to\R^{2n-1}$$f:N\to\R^{2n-1}$ a general position immersion which is an embedding on the closure of $N-D^n$$N-D^n$. Let $N_0 := N-B^n$$N_0 := N-B^n$ and $f_0:=f|_{N-B^n}$$f_0:=f|_{N-B^n}$. We omit $\Z_{(n)}$$\Z_{(n)}$-coefficients. Then $f_0^{-1}f(D^n)$$f_0^{-1}f(D^n)$ represents a 1-cycle in $N$$N$. The homology class of this 1-cycle equals minus the normal Stiefel-Whitney class $PD\overline{W}^{n-1}(N)\in H_1(N)$$PD\overline{W}^{n-1}(N)\in H_1(N)$. This follows because

$\displaystyle 0 \overset{(1)}= [f^{-1}f'(N)] \overset{(2)}= [f_0^{-1}f'(N_0)]+[f_0^{-1}f'(B^n)] \overset{(3)} = PD\overline{W}^{n-1}(N)+[f_0^{-1}f(D^n)].$ $$\displaystyle 0 \overset{(1)}= [f^{-1}f'(N)] \overset{(2)}= [f_0^{-1}f'(N_0)]+[f_0^{-1}f'(B^n)] \overset{(3)} = PD\overline{W}^{n-1}(N)+[f_0^{-1}f(D^n)].$$

Here

• $f':N\to\R^{2n-1}$$f':N\to\R^{2n-1}$ is the map formed by a general position small length vector field normal to $f$$f$, and non-zero over $f(B^n)$$f(B^n)$;

• equality (1) holds because $f'$$f'$ is homotopic to a map whose image is disjoint from $f$$f$;

• equality (2) holds because $f_0$$f_0$-preimages of both $f'(N_0)$$f'(N_0)$ and $f'B^n$$f'B^n$ are 1-cycles, and $f|_{B^n}$$f|_{B^n}$-preimage of $f'(N)$$f'(N)$ is a 1-cycle which is null-homologous;

• equality (3) holds because the Euler class of a bundle is dual to the homology class of zeroes of a general position section, and because $[f_0^{-1}f'(B^n)] = [f_0^{-1}f'(D^n)] \overset{(4)}= [f_0^{-1}f(D^n)]$$[f_0^{-1}f'(B^n)] = [f_0^{-1}f'(D^n)] \overset{(4)}= [f_0^{-1}f(D^n)]$;

• equality (4) holds because $f(D^n)$$f(D^n)$ is homologous relative to the boundary and outside $f(N_0)$$f(N_0)$ to the union of $f'(D^n)$$f'(D^n)$ and the the normal vectors at points of $f(\partial D^n)$$f(\partial D^n)$.

6 A generalization to highly-connected manifolds

In this section let $N$$N$ be a closed orientable homologically $k$$k$-connected $n$$n$-manifold, $k\ge0$$k\ge0$. Recall the unknotting theorem [Skopenkov2016c, Theorem 2.4] that all embeddings $N \to\Rr^m$$N \to\Rr^m$ are isotopic when $m\ge 2n-k+1$$m\ge 2n-k+1$ and $n\ge2k+2$$n\ge2k+2$. In this section we generalize Theorem 2.1 to a description of $E^{2n-k}(N)$$E^{2n-k}(N)$ and further to $E^m(N)$$E^m(N)$ for $m\ge2n-2k+1$$m\ge2n-2k+1$.

6.1 Examples

Some simple examples are the Hudson tori $\Hud_{m,n,p}(a), \Hud'_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$$\Hud_{m,n,p}(a), \Hud'_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$.

Example 6.1 (cf. [Skopenkov2010, Definition 1.4]). Assume that $N$$N$ is $k$$k$-connected and $n-k\ge3$$n-k\ge3$. Then for an embedding $f_0:N\to\Rr^{2n-k}$$f_0:N\to\Rr^{2n-k}$ and a class $a\in H_{k+1}(N;\Zz)$$a\in H_{k+1}(N;\Zz)$ one can construct an embedding $f_a:N\to\Rr^{2n-k}$$f_a:N\to\Rr^{2n-k}$ by linked embedded connected sum analogously to the case $k=0$$k=0$ presented in Example 4.1.

We have $W(f_a,f_0)=a$$W(f_a,f_0)=a$ for the Whitney invariant. Hence by Theorem 6.2 below this construction gives a free transitive action of $H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$$H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$ on $E^{2n-k}(N)$$E^{2n-k}(N)$, provided $n\ge k+3$$n\ge k+3$ or $n\ge2k+4$$n\ge2k+4$ in the PL or smooth categories, respectively.

The embedding $f_a$$f_a$ has an alternative construction using parametric connected sum [Skopenkov2014, Remark 18.a].

6.2 Classification

Theorem 6.2. Let $N$$N$ be a closed oriented homologically $k$$k$-connected $n$$n$-manifold, $k\ge0$$k\ge0$. Then the Whitney invariant

$\displaystyle W:E^{2n-k}(N)\to H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$ $$\displaystyle W:E^{2n-k}(N)\to H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$$

is a bijection, provided $n\ge2k+4$$n\ge2k+4$ in the smooth category or $n\ge k+3$$n\ge k+3$ in the PL category.

This was proved for $k$$k$-connected manifolds in the smooth category [Haefliger&Hirsch1963, Theorem 2.4], and in the PL category in [Weber1967], [Hudson1969, $\S$$\S$11], cf. [Boechat&Haefliger1970, Theorem 1.6], [Boechat1971, Theorem 4.2], [Vrabec1977, Theorems 1.1 and 1.2], [Adachi1993, $\S$$\S$7]. The proof actually used the homological $k$$k$-connectedness assumption (basically because the $k$$k$-connectedness was used to ensure high enough connectedness of the complement in $\Rr^m$$\Rr^m$ to the image of $N$$N$, by Alexander duality and simple connectedness of the complement, so homological $k$$k$-connectedness of $N$$N$ is sufficient).

For $k=0$$k=0$ Theorem 6.2 is covered by Theorem 2.1; for $k\ge1$$k\ge1$ it is not. For $k+3\le n\le2k+1$$k+3\le n\le2k+1$ the PL case of Theorem 6.2 gives nothing but the Zeeman Unknotting Spheres Theorem [Skopenkov2016c, Theorem 2.3]. For the case of knotted tori see [Skopenkov2016k, Theorem 3.1].

An inverse to the map $W$$W$ of Theorem 6.2 is given by Example 6.1.

Because of the existence of knotted spheres the analogues of Theorem 6.2 for $n=k+2$$n=k+2$ in the PL case, and for $n\le2k+3$$n\le2k+3$ in the smooth case are false. So for the smooth category, $n\le2k+3$$n\le2k+3$ and $N$$N$ closed connected, a classification of $E^{2n-k}(N)$$E^{2n-k}(N)$ is much harder: for 40 years the only known complete readily calculable classification results were for homology spheres $N$$N$. E.g. $E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)}$$E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)}$ for any $s\ge2$$s\ge2$ [Haefliger1966, Corollary 8.14], [Skopenkov2016s, Theorem 3.2]. The following result for $n=2k+3$$n=2k+3$ was obtained using the Boéchat-Haefliger formula for the smoothing obstruction [Boechat&Haefliger1970, Theorem 2.1], [Boechat1971, Theorem 5.1]. Using that formula one can define the higher-dimensional Kreck invariant [Skopenkov2008, $\S$$\S$4].

Theorem 6.3 [Skopenkov2008, Higher-dimensional Classification Theorem]. Let $N$$N$ be a closed orientable homologically $(2l-2)$$(2l-2)$-connected $(4l-1)$$(4l-1)$-manifold. Then the Whitney invariant

$\displaystyle W:E^{6l}_D(N)\to H_{2l-1}(N)$ $$\displaystyle W:E^{6l}_D(N)\to H_{2l-1}(N)$$

is surjective and for any $u\in H_{2l-1}(N)$$u\in H_{2l-1}(N)$ the Kreck invariant

$\displaystyle \eta_u:W^{-1}u\to\Zz_{d(u)}$ $$\displaystyle \eta_u:W^{-1}u\to\Zz_{d(u)}$$

is a 1-1 correspondence, where $d(u)$$d(u)$ is the divisibility of the projection of $u$$u$ to the free part of $H_1(N)$$H_1(N)$.

Recall that the divisibility of zero is zero and the divisibility of $x\in G-\{0\}$$x\in G-\{0\}$ is $\max\{d\in\Zz\ | \ \text{there is }x_1\in G: \ x=dx_1\}$$\max\{d\in\Zz\ | \ \text{there is }x_1\in G: \ x=dx_1\}$.

How does one describe $E^m(N)$$E^m(N)$ when $N$$N$ is not $(2n-m)$$(2n-m)$-connected? For general $N$$N$ see the sentence on $E^{2n-1}(N)$$E^{2n-1}(N)$ at the end of $\S$$\S$2. We can say more as the connectivity $k$$k$ of $N$$N$ increases. Some estimations of $E^{2n-k-1}(N)$$E^{2n-k-1}(N)$ for a closed $k$$k$-connected $n$$n$-manifold $N$$N$ are presented in [Skopenkov2010]. For $k>1$$k>1$ one can go even further:

Theorem 6.4 [Becker&Glover1971, Corollary 1.3]. Let $N$$N$ be a closed $k$$k$-connected $n$$n$-manifold embeddable into $\Rr^m$$\Rr^m$, $m\ge2n-2k+1$$m\ge2n-2k+1$ and $2m\ge 3n+4$$2m\ge 3n+4$. Then there is a 1-1 correspondence

$\displaystyle E^m(N)\to [N_0, V_{m,n+1}].$ $$\displaystyle E^m(N)\to [N_0, V_{m,n+1}].$$

The 1-1 correspondence is constructed using `the Haefliger-Wu invariant' involving the configuration space of distinct pairs [Skopenkov2006, $\S$$\S$5]. For $k=0$$k=0$ Theorem 6.4 is the same as General Position Theorem [Skopenkov2016c, Theorem 2.1] (because $V_{2n+1,n+1}$$V_{2n+1,n+1}$ is $(n-1)$$(n-1)$-connected). For $k=1$$k=1$ Theorem 6.4 is covered by Theorem 6.2; for $k\ge2$$k\ge2$ it is not. For application to knotted tori see [Skopenkov2016k, Theorem 5.1]. For generalization to arbitrary manifolds see survey [Skopenkov2006, $\S$$\S$5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Observe that in Theorem 6.4 $V_{m,n+1}$$V_{m,n+1}$ can be replaced by $V_{M,M+n-m+1}$$V_{M,M+n-m+1}$ for any $M>n$$M>n$.

7 An orientation on the self-intersection set

Let $f:N\to\Rr^m$$f:N\to\Rr^m$ be a smooth map from an oriented $n$$n$-manifold $N$$N$ where $m\ge n+2$$m\ge n+2$. We assume that the closure

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${\rm Cl}\Sigma(f)$ of the self-intersection set of $f$$f$ has codimension 2 singularities, i.e., there is

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$P\subset{\rm Cl}\Sigma(f)$ such that

• both $P$$P$ and

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  ${\rm Cl}\Sigma(f)$ are subpolyhedra of some triangulation of $N$$N$,
• we have $\dim P\le\dim\Sigma(f)-2$$\dim P\le\dim\Sigma(f)-2$ and
• $\Sigma(f)-P$$\Sigma(f)-P$ is an open manifold consisting of self-transverse double points of $f$$f$.

Definition 7.1 (A canonical orientation on $\Sigma(f)-P$$\Sigma(f)-P$). Take points $x,y\in N$$x,y\in N$ away from $P$$P$ and such that $fx=fy$$fx=fy$. Then a $(2n-m)$$(2n-m)$-base $\xi_x$$\xi_x$ tangent to $\Sigma(f)-P$$\Sigma(f)-P$ at $x$$x$ gives a $(2n-m)$$(2n-m)$-base $\xi_y:=df_y^{-1}df_x(\xi_x)$$\xi_y:=df_y^{-1}df_x(\xi_x)$ tangent to $\Sigma(f)-P$$\Sigma(f)-P$ at $y$$y$. Since $N$$N$ is oriented, we can take positive $(m-n)$$(m-n)$-bases $\eta_x$$\eta_x$ and $\eta_y$$\eta_y$ at $x$$x$ and $y$$y$ normal to $\xi_x$$\xi_x$ and to $\xi_y$$\xi_y$. If the base $(df_x(\xi_x),df_x(\eta_x),df_y(\eta_y))$$(df_x(\xi_x),df_x(\eta_x),df_y(\eta_y))$ of $\Rr^m$$\Rr^m$ is positive, then call the base $\xi_x$$\xi_x$ positive. This is well-defined because a change of the sign of $\xi_x$$\xi_x$ forces changes of the signs of $\xi_y,\eta_x$$\xi_y,\eta_x$ and $\eta_y$$\eta_y$.

Remark 7.2 (Properties of the orientation just defined on $\Sigma(f) - P$$\Sigma(f) - P$)..

1. A change of the orientation of $N$$N$ forces changes of the signs of $\eta_x$$\eta_x$ and $\eta_y$$\eta_y$ and so does not change the orientation of $\Sigma(f)-P$$\Sigma(f)-P$.
2. The orientation on $\Sigma(f)-P$$\Sigma(f)-P$ need not extend to

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   ${\rm Cl}\Sigma(f)$: take the smooth cone $f:D^3\to\Rr^5$$f:D^3\to\Rr^5$ over a general position map $g:S^2\to\Rr^4$$g:S^2\to\Rr^4$ having only two transverse self-intersection points, where the smooth cone is defined by $f(tx):=(g(x)\sin(\pi t/2),\cos(\pi t/2))$$f(tx):=(g(x)\sin(\pi t/2),\cos(\pi t/2))$, for $x\in S^2$$x\in S^2$ and $t\in[0,1]$$t\in[0,1]$.
3. The orientation on $\Sigma(f)-P$$\Sigma(f)-P$ extends to

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   ${\rm Cl}\Sigma(f)$ if $m-n$$m-n$ is odd [Hudson1969, Lemma 11.4].

Remark 7.3 (A canonical orientation on $f\Sigma(f)-fP$$f\Sigma(f)-fP$ for $m-n$$m-n$ even). This remark is added as a complement for Definition 7.1 but is not needed for the definition of the Whitney invariant.

Take a $(2n-m)$$(2n-m)$-base $\xi$$\xi$ at a point $x\in f\Sigma(f)-fP$$x\in f\Sigma(f)-fP$. Since $N$$N$ is oriented, we can take a positive $(m-n)$$(m-n)$-base $\eta_+$$\eta_+$ normal to $f\Sigma(f)$$f\Sigma(f)$ in one sheet of $fN$$fN$. Analogously construct an $(m-n)$$(m-n)$-base $\eta_-$$\eta_-$ for the other sheet of $fN$$fN$. Since $m-n$$m-n$ is even, the orientation of the base $(\xi,\eta_+,\eta_-)$$(\xi,\eta_+,\eta_-)$ of $\Rr^m$$\Rr^m$ does not depend on choosing the first and the other sheet of $fN$$fN$ at $x$$x$. If the base $(\xi,\eta_+,\eta_-)$$(\xi,\eta_+,\eta_-)$ is positive, then call the base $\xi$$\xi$ positive. This is well-defined because a change of the sign of $\xi$$\xi$ forces changes of the signs of $\eta_+,\eta_-$$\eta_+,\eta_-$ and so of $(\xi,\eta_+,\eta_-)$$(\xi,\eta_+,\eta_-)$.

We remark that a change of the orientation of $N$$N$ forces changes of the signs of $\eta_+,\eta_-$$\eta_+,\eta_-$ and so does not change the orientation of $f\Sigma(f)-fP$$f\Sigma(f)-fP$.

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\times S^{n-1}$ and $\overline a$ the `standard inclusion' $\{-1\}\times S^{n-1}\to S^{n-1}=\partial D^n\subset D^n$ on the other component. standard $S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$--> This map gives an embedding $$S^0\times S^{n-1}\xrightarrow{\widetilde a\times {\rm pr}_2}S^{n-1}\times S^{n-1}\subset D^{n+1}\times S^{n-1}\overset{{\rm i}}\to\Rr^{2n}.$$ See Figure 1. The image of $\widetilde a\times{\rm pr}_2$ is the union of the graphs of the maps $\overline a$ and $-\overline a$. For any $t\in S^{n-1}$ the disk ${\rm i}(D^{n+1}\times t)$ intersects the image at two points lying in ${\rm i}(D^n\times t)$, i.e., at the image of an embedding $S^0\times t\to {\rm i}(D^n\times t)$. The embedding $\Hud_n'(a)$ is obtained by extending the latter embeddings to embeddings $S^1\times t\to {\rm i}(D^{n+1}\times t)$ for all $t$. See Figure 2. [[Image:construction_hudson_torus.jpg|thumb|250px|Figure 2: To the construction of the Hudson torus]] {{endthm}} {{beginthm|Remark}}\label{rhe} (a) The analogue of Proposition \ref{pr3} for $\Hud_n$ replaced to $\Hud_n'$ holds, with an analogous proof. (b) The embeddings $\Hud_n(a)$ and $\Hud_n'(a)$ are smoothly isotopic for $n\ge4$ and are PS isotopic for $n\ge3$ \cite[commutativity of the left upper square in the Restriction Lemma 5.2]{Skopenkov2006a}, \cite[Lemma 2.15.c]{Skopenkov2015a} (see \cite[Remark 1.2]{Skopenkov2016f}). This follows by calculation of [[#The_Whitney_invariant|the Whitney invariant]] (Remark \ref{re5}.d below). It would be interesting to know if they are smoothly isotopic for $n=3$. It would be interesting to know if they are piecewise smoothly isotopic for $n=2$. (b') Although construction of $\Hud_n(a)$ and $\Hud_n'(a)$ is explicit, no explicit construction of an isotopy between them is known. Proof from (b) of their being isotopic is by calculating invariants, and using completeness result for the invariants. This proof is fairly constructive, so in principle it does give a specific isotopy. However, the construction is so complicated that one would not call such an isotopy explicit. This is analogous to the Smale `turning sphere inside out' theorem (although the Smale theorem concerns regular homotopy of immersions not isotopy of embeddings). In higher dimensions, a proof that ''any'' embedding is isotopic to some of the constructed embeddings is even less likely to use an explicit construction. The only proofs we have go through calculating the invariants and using classification results. (c) For $n=2$ Example \ref{dh2n} gives what we call the 'left' Hudson torus. The 'right' Hudson torus is constructed analogously. It is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of $S^1\times S^1$. The right and the left Hudson tori are not isotopic by Remark \ref{re5}.d below. (d) Analogously one constructs the Hudson torus $\Hud_{n,p}(a):S^p\times S^{n-p}\to\Rr^{2n-p+1}$ for $a\in\Zz$ and $n>p\ge0$ or, more generally, $\Hud_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$ for $a\in\pi_n(S^{m-n+p-1})$ and $m>n>p\ge0$. There are versions $\Hud'_{m,n,p}(a)$ of these constructions corresponding to Definition \ref{dh2n}. For $p=0$ this corresponds to [[High_codimension_links#Examples|the Zeeman map]] \cite[Definition 2.2]{Skopenkov2016h} and its composition with 'the unframed second Kirby move' \cite[$\S.3]{Skopenkov2015a}. It would be interesting to know if the links $\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$ are isotopic, cf. \cite[Remark 2.7.b]{Skopenkov2015a}. These constructions can be [[Knotted_tori#Examples|further generalized]] \cite{Skopenkov2016k}. {{endthm}} {{beginthm|Example}}\label{e:lato} Define a smooth embedding $$l:S^1\times S^{n-1}\to \C^n=\R^{2n}\quad\text{by}\quad l(z,x) := \left(1+\frac z2\right)x.$$ Here $S^1\subset\R^2=\C$ and $S^{n-1}\subset\R^n\subset\C^n$. This embedding is known to be ''Lagrangian''; see more in \cite{Nemirovski2024}. {{endthm}} ==Action by linked embedded connected sum== ; \label{s:action} In this section we generalize the construction of the Hudson torus $\Hud(a)$. Let $N$ be a closed connected oriented $n$-manifold. We work in the smooth category which we omit. Apparently analogous results hold for $n\ge3$ in the PL and PS categories (see \cite[Remark 1.2]{Skopenkov2016f}). {{beginthm|Example}}\label{d:lecs} For any $n\ge4$, an embedding $f_0:N\to\Rr^{2n}$ and $a\in H_1(N;\Zz)$, we shall construct an embedding $f_a:N\to\Rr^{2n}$. This embedding is said to be obtained by linked embedded connected sum of $f_0$ with an $n$-sphere representing the `homology Alexander dual' $A:=\widehat{A_{f_0}}a\in H_n(C_{f_0})$ of $a$ (defined in \cite[Alexander Duality Lemma 4.6]{Skopenkov2005}). Represent $a$ by an embedding $a:S^1\to N$. By definition, the class $A$ is represented by properly oriented $\nu_{f_0}^{-1}a(S^1)$. Since any orientable bundle over $S^1$ is trivial, $\nu_{f_0}^{-1}a(S^1)\cong S^1\times S^{n-1}$. Take an embedding $g:S^1\times S^{n-1}\to C_{f_0}$ whose image is $\nu_{f_0}^{-1}a(S^1)$ and which represents $A$. By ''embedded surgery'' on $S^1\times1_{n-1}\subset S^1\times S^{n-1}$ we obtain an embedding $g_1:S^n\to C_{f_0}$ representing $A$ (see details in Proposition \ref{p:es} below). Define $f_a$ to be the linked embedded connected sum of $f_0$ and $g_1$, along some arc joining their images. {{endthm}} {{beginthm|Proposition|(Embedded surgery)}}\label{p:es} For any $n\ge3$, a neighborhood $U$ of a codimension at least 3 subpolyhedron in $\Rr^{2n}$ and an embedding $g:S^1\times S^{n-1}\to\Rr^{2n}-U$ there is an embedding $g_1:S^n\to\Rr^{2n}-U$ homologous to $g$. {{endthm}} ''Proof''. Take a vector field on $g(S^1\times1_{n-1})$ normal to $g(S^1\times S^{n-1})$. Extend $g|_{S^1\times1_{n-1}}$ along this vector field to a map $\overline b:D^2\to\Rr^{2n}$. Since n>4$ and $U$ is a neighborhood $U$ of a codimension at least 3 subpolyhedron, by general position we may assume that $\overline b$ is an embedding and that $\overline b({\rm Int}D^2)$ misses $U\cup g(S^1\times S^{n-1})$. Since $n-1>1$, we have $\pi_1(V_{2n-2,n-1})=0$. Hence the standard $(n-1)$-framing of $S^1\times1_{n-1}$ in $S^1\times S^{n-1}$ extends to an $(n-1)$-framing on $\overline b(D^2)$ in $\Rr^{2n}$. Thus $\overline b$ extends to an embedding $$ \widehat b:D^2\times D^{n-1}\to\R^{2n}-U\quad\text{such that}\quad \widehat b(\partial D^2\times D^{n-1})\subset g(S^1\times S^{n-1}). $$ Take an embedding $g_1:S^n\to\R^{2n}-U$ such that $$ g_1(S^n)\ =\ g(S^1\times S^{n-1})-\widehat b(\partial D^2\times{\rm Int}D^{n-1}) \bigcup\limits_{\widehat b(\partial D^2\times\partial D^{n-1})} \widehat b(D^2\times\partial D^{n-1})\ \cong\ S^n. $$ with proper orientation so that $g_1$ is homologous to $g$. QED The isotopy class of the embedding $f_a$ is independent of the choises in the construction. The independence of the arc and of the maps $g,g_1$ follows by $n\ge3$ and by Proposition \ref{act} below, respectively. By Definition \ref{dw1} of [[#The_Whitney_invariant|the Whitney invariant]], $W(f_a,f_0)$ is $a$ for $n\ge3$ odd and $a \mod2$ for $n\ge4$ even. Thus by Theorem \ref{th4}.a for $n\ge4$ all isotopy classes of embeddings $N\to\Rr^{2n}$ can be obtained from any chosen embedding $f_0$ by the above construction. {{beginthm|Proposition}}\label{act} For any $n\ge4$ both the linked embedded connected sum and [[Parametric_connected_sum#Applications|parametric connected sum]] (introduced in \cite{Skopenkov2006a}, \cite{Skopenkov2015a}) define free transitive actions of $H_1(N;\Zz_{\varepsilon(n-1)})$ on $E^{2n}(N)$. {{endthm}} This follows by Theorem \ref{th4}.a and by \cite[Remark 18.a]{Skopenkov2014}. == The Whitney invariant == ; \label{s:whitney} Let $N$ be a closed $n$-manifold. Take an embedding $f_0:N\to\Rr^m$. Fix an orientation on $\Rr^m$. For any other embedding $f \colon N \to \Rr^m$ we define ''the Whitney invariant'' $$W(f, f_0)=W_{f_0}(f)=W(f)\in H_{2n-m+1}(N;\Zz_N).$$ Here the coefficients $\Zz_N$ are $\Zz$ if $N$ is oriented and $m-n$ is odd, and are $\Zz_2$ otherwise. Roughly speaking, $W(f):=[{\rm Cl}\Sigma(H)]$ is defined as the homology class of the closure of the self-intersection set of a general position homotopy $H$ between $f$ and $f_0$. This is formalized in Definition \ref{dw2} in the smooth category, following \cite{Skopenkov2010}, see also \cite{Haefliger&Hirsch1963}. The definition in the PL category is analogous \cite[$\S]{Hudson1969}, \cite[p. 145]{Vrabec1977}, \cite[$\S.4 `The Whitney invariant']{Skopenkov2006}. We begin by presenting a simpler definition, Definition \ref{dw1}, for a particular case. For Theorem \ref{th4} only the case $m=2n$ is required. {{beginthm|Definition}}\label{dw1} Assume that $N$ is $(2n-m)$-connected and m\ge3n+3$. Then by \cite[Theorem 3.1.b]{Haefliger&Hirsch1963} restrictions of $f$ and $f_0$ to $N_0$ are isotopic, cf. \cite[Lemma 2.2]{Takase2006}. (Here is sketch of an argument. Using the Smale-Hirsch classification of immersions we obtain that restrictions of $f$ and $f_0$ to $N_0$ are `regular homotopic', see \cite[Definition 2.7]{Koschorke2013}. Since $N$ is $(2n-m)$-connected, $N_0$ retracts to an $(m-n-1)$-dimensional polyhedron. Therefore these restrictions are isotopic.) So we can make an isotopy of $f$ and assume that $f=f_0$ on $N_0$. Take a general position homotopy $F:B^n\times I\to\Rr^m$ relative to $\partial B^n$ between the restrictions of $f$ and $f_0$ to $B^n$. Let $f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$ (`the intersection of this homotopy with $f(N-B^n)$'). Since $n+2(n+1)<2m$, by general position ${\rm Cl}(f\cap F)$ is a compact $(2n+1-m)$-manifold whose boundary is contained in $\partial N_0$. So $f\cap F$ carries a homology class with $\Zz_2$ coefficients. If $m-n$ is odd and $N$ is oriented, then $f\cap F$ has a natural orientation defined below, and so carries a homology class with $\Zz$ coefficients. Define $W(f)$ to be the homology class: $$ W(f):=[{\rm Cl}(f\cap F)]\in H_{2n-m+1}(N_0,\partial N_0;\Zz_N)\cong H_{2n-m+1}(N;\Zz_N). $$ The orientation on $f\cap F$ (extendable to ${\rm Cl}(f\cap F)$) is defined (for $m-n$ odd and $N$ is oriented) as follows (cf. Remark \ref{orifs}). For any point $x_f\in f\cap F$ take a base $\xi_f$ at $x_f$ tangent to $f\cap F$. Complete this base to a positive base $(\xi_f,\eta_f)$ tangent to $N$. Since $n+2(n+1)<2m$, by general position there is a unique point $x_F\in B^n\times I$ such that $Fx_F=fx_f$. The tangent base $\xi_f$ at $x_f$ thus gives a base $\xi_F$ at $x_F$ tangent to $B^n\times I$ such that $df(x_f)\xi_f=dF(x_F)\xi_F$. Complete this base $\xi_F$ to a positive base $(\xi_F,\eta_F)$ tangent to $B^n\times I$, where the orientation on $B^n$ comes from $N$. The union $\zeta=(df(x_f)\xi_f,df(x_f)\eta_f,dF(x_F)\eta_F)$ of the images of the constructed bases is a base at $fx_f=Fx_F$ of $\Rr^m$. If $\zeta$ is positive, then call the tangent base $\xi_f$ of $f\cap F$ `positive'. Since a change of the orientation on $f\cap F$ forces a change of the orientation of $\zeta$, this condition indeed defines an orientation on $f\cap F$. {{endthm}} {{beginthm|Definition}}\label{dw2} Assume that $m\ge n+2$. Take a general position homotopy $H:N\times I\to\Rr^m\times I$ between $f_0$ and $f$. The closure ${\rm Cl}\Sigma(H)$ of the self-intersection set carries a cycle mod 2. If $N$ is oriented and $m-n$ is odd, the closure also carries an integer cycle. See \cite[$\S]{Hudson1967}, \cite[$\S.3 `The Whitney obstruction']{Skopenkov2006}. Let us informally explain these facts. For m\ge3n+2$ by general position the closure ${\rm Cl}\Sigma(H)$ can be assumed to be a submanifold. In general, since $m\ge n+2$, by general position the closure has codimension 2 singularities, as defined in $\S$\ref{s:orien}. So the closure carries a cycle mod 2. When $m-n$ is odd the closure also has a [[#An_orientation_on_the_self-intersection_set|canonical orientation]] (see Definition \ref{oris} and Remark \ref{remoris}), so the closure carries an integer cycle. Define the Whitney invariant to be the homology class: $$ W(f):=[{\rm Cl}\Sigma(H)]\in H_{2n-m+1}(N\times I;\Zz_N)\cong H_{2n-m+1}(N;\Zz_N). $$ Clearly, $W(f) = W(f')$ if $f$ is isotopic to $f'$. Hence the Whitney invariant defines a map $$W:E^m(N)\to H_{2n-m+1}(N;\Zz_N),\quad [f] \mapsto W(f).$$ {{endthm}} Clearly, $W(f_0)=0$ (for both definitions). The definition of $W$ depends on the choice of $f_0$, but we write $W$ not $W_{f_0}$ for brevity. {{beginthm|Remark}}\label{re5} (a) The Whitney invariant is well-defined by Definition \ref{dw2}, i.e. is independent of the choice of a general position homotopy $H:N\times I\to\Rr^m\times I$ from $f_0$ to $f$. This follows from the equality $[{\rm Cl}\Sigma(H_0)]−[{\rm Cl}\Sigma(H_1)] = \partial [{\rm Cl}\Sigma(H_{01})]$ for a general position homotopy $H_{01}:N\times I\times I\to\Rr^m\times I\times I$ between general position homotopies $H_0,H_1:N\times I\to\Rr^m\times I$ from $f_0$ to $f$. See details in \cite[$\S]{Hudson1969}. (b) Definition \ref{dw1} is a particular case of Definition \ref{dw2}. (Indeed, if $f=f_0$ on $N_0$, we can take $H$ to be fixed on $N_0$. See details in \cite[Difference Lemma 2.4]{Skopenkov2010}.) Hence the Whitney invariant is well-defined by Definition \ref{dw1}, i.e. independent of the choice of $F$ and of the isotopy making $f=f_0$ outside $B^n$. (c) The class $W(f)$ is independent of the choice of the orientation on $N$ (because a change of the orientation on $N$ forces a change of the orientation on $f\cap F$ or on ${\rm Cl}\Sigma(H)$). For the reflection $\sigma:\Rr^m\to\Rr^m$ with respect to a hyperplane we have $W(\sigma\circ f)=-W(f)$ (because a change of the orientation on $\Rr^m$ forces a change of the orientation on $f\cap F$ or on ${\rm Cl}\Sigma(H)$; for Definition \ref{dw1} also observe that we may assume that $f=f_0=\sigma\circ f$ on $N_0$). (d) For the [[#Hudson tori|Hudson tori]] $W(\Hud_n(a))=W(\Hud'_n(a))$ is $a$ or $a\mod2$ for $n\ge3$, and $W(\Hud'_2(a))=(a\mod2,0)$. For $\Hud_n(a)$ this is clear by Definition \ref{dw1}. For $\Hud'_n(a)$ and $n\ge3$ this was proved in \cite{Hudson1963} (with a different but equivalent definition of the Whitney invariant; using and proving a particular case of Remark \ref{re5}.f). For $\Hud'_2(a)$ the proof is analogous. (e) $W(f\#g)=W(f)$ for any pair of embeddings $f:N\to\Rr^m$ and $g:S^n\to\Rr^m$. This is clear by Definition \ref{dw1} because $W(f\#g)-W(f)=W(f\#g,f_0)-W(f,f_0)=W(f\#g,f)=0$. Let us prove the latter equality. Take the identical isotopy $H_f$ of $f$ and a general position homotopy $H_g$ between $g$ and the standard embedding. Then the boundary connected sum $H_f\sharp H_g$ is a general position homotopy between $f\#g$ and an embedding isotopic to $f$. The cycle ${\rm Cl}\Sigma(H_f\sharp H_g)={\rm Cl}\Sigma H_g$ is null-homologous in $S^n$ and hence in $N\cong N\#S^n$; cf. \cite[Addendum to the Classification Theorem]{Skopenkov2008}. (f) For $m=2n+1$ and $N=S^n\sqcup S^n$ the Whitney invariant equals to the pair of [[High_codimension_links#The_linking_coefficient|''linking coefficients'']] \cite[$\S]{Skopenkov2016h}. (g) The Whitney invariant need not be a bijection for $m<2n$. This is seen, for example, by applying Theorem \ref{begl} below in case of [[Knotted_tori#Further_classification|knotted tori]] \cite[Theorem 5.1]{Skopenkov2016k}) or by taking $n$ even, $N$ non-orientable, $m=2n$ and applying by Theorem \ref{th4}.b. (h) The paper \cite[$\S]{Haefliger&Hirsch1963} gives yet another equivalent definition of the Whitney invariant, and proves its injectivity under certain restrictions (the Haefliger-Hirsch $h$-principle, see Theorem \ref{hico}). (i) Let $N$ be a closed orientable $n$-manifold. For two embeddings $f,g:N\to\R^{2n-1}$ the invariant defined in \cite[$\S]{Haefliger&Hirsch1963} equals plus or minus their Whitney invariant $W(f,g)$ from Definition \ref{dw1}. This is proved analogously to the following version. Let $D^n$ and $B^n$ be closed $n$-balls in $N$ such that $D^n\subset{\rm Int}B^n$. Let $f:N\to\R^{2n-1}$ a general position immersion which is an embedding on the closure of $N-D^n$. Let $N_0 := N-B^n$ and $f_0:=f|_{N-B^n}$. We omit $\Z_{(n)}$-coefficients. Then $f_0^{-1}f(D^n)$ represents a 1-cycle in $N$. The homology class of this 1-cycle equals minus the normal Stiefel-Whitney class $PD\overline{W}^{n-1}(N)\in H_1(N)$. This follows because $$N$ is a connected manifold of dimension $n>1$$n>1$, and $m \ge2n+1$$m \ge2n+1$, then every two embeddings $N \to\Rr^m$$N \to\Rr^m$ are isotopic [Skopenkov2016c, Theorem 3.2], [Skopenkov2006, Theorem 2.5]. In this page we summarize the situation for $m=2n\ge6$$m=2n\ge6$ and $N$$N$ is a connected, as well as in some more general situations. For the classification of embeddings of some disconnected manifolds see [Skopenkov2016h].

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$$\S$1, $\S$$\S$3]. Denote $1_n:=(1,0,\ldots,0)\in S^n$$1_n:=(1,0,\ldots,0)\in S^n$.

2 Classification

For the next theorem, the Whitney invariant $W$$W$ is defined in $\S$$\S$5 below.

Theorem 2.1. Assume that $N$$N$ is a closed connected $n$$n$-manifold, and either $n\ge4$$n\ge4$ or $n=3$$n=3$ and we are in the PL category.

(a) If $N$$N$ is oriented, the Whitney invariant,

$\displaystyle W:E^{2n}(N)\to H_1(N;\Zz_{\varepsilon(n-1)}),$ $$\displaystyle W:E^{2n}(N)\to H_1(N;\Zz_{\varepsilon(n-1)}),$$

is a 1-1 correspondence.

(b) If $N$$N$ is non-orientable, then there is a 1-1 correspondence

$\displaystyle E^{2n}(N)\to \begin{cases} H_1(N;\Zz_2) & n\text{ is odd}\\ \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s & n\text{ is even}\end{cases}.$ $$\displaystyle E^{2n}(N)\to \begin{cases} H_1(N;\Zz_2) & n\text{ is odd}\\ \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s & n\text{ is even}\end{cases}.$$

Remark 2.2 (Comments on the proof). Part (a) is proved in [Haefliger&Hirsch1963, Theorem 2.4] in the smooth category, and in [Weber1967, Theorem 4' in $\S$$\S$2], [Hudson1969, $\S$$\S$11], [Vrabec1977, Theorems 1.1 and 1.2] in the PL category, see also [Haefliger1962b, 1.3.e], [Haefliger1963], [Bausum1975, Theorem 43].

Part (b) is proved in [Bausum1975, Theorem 43] in the smooth category. By [Weber1967, Theorems 1 and 1' in $\S$$\S$2], [Skopenkov1997, Theorem 1.1.c] the proof works also in the PL category.

In Part (b) we replaced the kernel $\ker Sq^1$$\ker Sq^1$ from [Bausum1975, Theorem 43] by $\Zz_2^{s-1}$$\Zz_2^{s-1}$. This is possible because, as a specialist could see, $Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$$Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$ is given by multiplication with the first Stefel-Whitney class $w_1(N)$$w_1(N)$ (which equals to the first Wu class $v_1(N)$$v_1(N)$ [Milnor&Stasheff1974, Theorem 11.4]). Since $N$$N$ is non-orientable, $w_1(N)\neq 0$$w_1(N)\neq 0$. So by Poincaré duality, $\ker Sq^1 \cong \Zz_2^{s-1}$$\ker Sq^1 \cong \Zz_2^{s-1}$.

The 1-1 correspondence from (b) can presumably be defined as a (generalized) Whitney invariant, see [Vrabec1977], but the proof used 'the Haefliger-Wu invariant' whose definition can be found e.g. in [Skopenkov2006, $\S$$\S$5]. It would be interesting to check if part (b) is equivalent to different forms of description of $E^{2n}(N)$$E^{2n}(N)$ [Haefliger1962b, 1.3.e], [Haefliger1963], [Vrabec1977, Theorem 1.1].

The classification of smooth embeddings of 3-manifolds in $\Rr^6$$\Rr^6$ is more complicated, see Theorem 6.3 below for $l=1$$l=1$ or [Skopenkov2016t].

Concerning embeddings of connected $n$$n$-manifolds in $\Rr^{2n-1}$$\Rr^{2n-1}$ see [Yasui1984] for $n\ge5$$n\ge5$, [Skopenkov2016f] for $n=4$$n=4$, and [Saeki1999], [Skopenkov2010], [Tonkonog2010] for manifolds with boundary.

Theorem 2.1 is generalized to a description of $E^{2n-k}(N)$$E^{2n-k}(N)$ for closed $k$$k$-connected $n$$n$-manifolds $N$$N$, see Theorem 6.2.

3 Hudson and Lagrangian tori

Together with the Haefliger knotted sphere [Skopenkov2016t, Example 2.1], [Skopenkov2006, Example 3.4], the examples of Hudson tori presented below were the first examples of non-isotopic embeddings in codimension greater than 2. (Hudson's construction [Hudson1963] was not as explicit as those below.) Abbreviate

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${\rm i}_{2n,n-1}$ to just

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${\rm i}$.

Example 3.1. Let us construct, for any $a\in\Zz$$a\in\Zz$ and $n\ge2$$n\ge2$, a smooth embedding

$\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.$ $$\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.$$

We start with the cases $a=0,1$$a=0,1$.

Take the standard inclusion $\partial D^2\subset\partial D^{n+1}$$\partial D^2\subset\partial D^{n+1}$. The 'standard embedding' $\Hud_n(0)$$\Hud_n(0)$ is given by the standard inclusions

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$$\displaystyle \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\overset{{\rm i}}\to\Rr^{2n}.$$

Define the `standard embedding'

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$2{\rm i}:2D^{n+1}\times S^{n-1}\to\Rr^{2n}$ analogously to

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${\rm i}$, where $2$$2$ means homothety with coefficient 2.

Take the embedding $g_1$$g_1$ given by

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$$\displaystyle S^n=2\partial D^{n+1}\times1_{n-1}\subset 2D^{n+1}\times S^{n-1}\overset{2{\rm i}}\to\Rr^{2n}.$$

The segment

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$2{\rm i}([1,2]1_n\times1_{n-1})$ joins the images of $\Hud_n(0)$$\Hud_n(0)$ and $g_1$$g_1$; the interior of this segment misses the images. Let $\Hud_n(1)$$\Hud_n(1)$ be the linked embedded connected sum of $\Hud_n(0)$$\Hud_n(0)$ and $g_1$$g_1$ along this segment, compatible with the orientation, cf. [Avvakumov2017, $\S$$\S$1.5]. (Here 'linked' means that the images of the embeddings are not contained in disjoint cubes, unlike for the unlinked embedded connected sum [Skopenkov2016c, $\S$$\S$5].)

For $a\in\Zz$$a\in\Zz$ we repeat the above construction of $g_1$$g_1$ replacing $2\partial D^{n+1}\times1_{n-1}$$2\partial D^{n+1}\times1_{n-1}$ by $|a|$$|a|$ copies $(1+\frac1k)\partial D^{n+1}\times1_{n-1}$$(1+\frac1k)\partial D^{n+1}\times1_{n-1}$ of $S^n$$S^n$, $k=1,\ldots,|a|$$k=1,\ldots,|a|$. The copies are outside $D^{n+1}\times S^{n-1}$$D^{n+1}\times S^{n-1}$ and are `parallel' to $\partial D^{n+1}\times1_{n-1}$$\partial D^{n+1}\times1_{n-1}$. The copies have the standard orientation for $a>0$$a>0$ or the opposite orientation for $a<0$$a<0$. Then we make embedded connected sum along natural segments joining every $k$$k$-th copy to the $(k+1)$$(k+1)$-th copy. We obtain an embedding $g_a:S^n\to\Rr^{2n}$$g_a:S^n\to\Rr^{2n}$ which has disjoint images with $\Hud_n(0)$$\Hud_n(0)$. Let $\Hud_n(a)$$\Hud_n(a)$ be the linked embedded connected sum of $\Hud_n(0)$$\Hud_n(0)$ and $g_a$$g_a$.

The original motivation for Hudson was that $\Hud_n(1)$$\Hud_n(1)$ is not isotopic to $\Hud_n(0)$$\Hud_n(0)$ for any $n\ge3$$n\ge3$ (this is a particular case of Proposition 3.2 below). One might guess that $\Hud_n(a)$$\Hud_n(a)$ is not isotopic to $\Hud_n(a')$$\Hud_n(a')$ for $a\ne a'$$a\ne a'$ and that a $\Zz$$\Zz$-valued invariant of $E^{2n}(S^1 \times S^{n-1})$$E^{2n}(S^1 \times S^{n-1})$ can be defined by the homotopy class of the map

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$$\displaystyle S^n\overset g\to S^{2n}-{\rm i}(D^{n+1}\times S^{n-1})\sim S^{2n}-S^{n-1}\sim S^n \quad\text{which is}\quad a\in\pi_n(S^n)\cong\Zz.$$

However, this is only true for $n$$n$ odd.

Proposition 3.2. For $n\ge3$$n\ge3$ odd $\Hud_n(a)$$\Hud_n(a)$ is isotopic to $\Hud_n(a')$$\Hud_n(a')$ if and only if $a=a'$$a=a'$.

For $n\ge4$$n\ge4$ even $\Hud_n(a)$$\Hud_n(a)$ is isotopic to $\Hud_n(a')$$\Hud_n(a')$ if and only if $a\equiv a'\text{ mod}2$$a\equiv a'\text{ mod}2$.

Proposition 3.2 follows by calculation of the Whitney invariant (Remark 5.3.d below) and, for $n$$n$ even, by Theorem 2.1. This proposition holds with the same proof in the piecewise smooth category, whose definition is recalled in [Skopenkov2016f, Remark 1.1]). Proposition 3.2 also holds in the PL category (with an analogous construction of $\Hud_n(a)$$\Hud_n(a)$ for the PL category). It would be interesting to find an explicit construction of an isotopy between $\Hud_{2k}(a)$$\Hud_{2k}(a)$ and $\Hud_{2k}(a+2)$$\Hud_{2k}(a+2)$, cf. [Vrabec1977, $\S$$\S$5]. Analogously, $\Hud_2(a)$$\Hud_2(a)$ is not isotopic to $\Hud_2(a')$$\Hud_2(a')$ if $a\not\equiv a'\text{ mod}2$$a\not\equiv a'\text{ mod}2$. It would be interesting to know if the converse holds, e.g. is $\Hud_2(0)$$\Hud_2(0)$ (PS or smoothly) isotopic to $\Hud_2(2)$$\Hud_2(2)$.

Example 3.3. Take any $a\in\Zz$$a\in\Zz$. Take a map $\overline a:S^{n-1}\to S^{n-1}$$\overline a:S^{n-1}\to S^{n-1}$ of degree $a$$a$ (so we can take $\overline 1=\id$$\overline 1=\id$). Recall that $D^{n+1}=\{(y,x)\in D^n\times D^1\ :\ |y|^2+|x|^2\le1\}$$D^{n+1}=\{(y,x)\in D^n\times D^1\ :\ |y|^2+|x|^2\le1\}$. Define the smooth embedding $\Hud_n'(a)$$\Hud_n'(a)$ to be the composition

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$$\displaystyle S^1\times S^{n-1}\overset{\widehat a}\to D^{n+1}\times S^{n-1}\overset{\rm i}\to\Rr^{2n} \quad\text{where}\quad \widehat a(e^{i\theta},y):=(\overline a(y)\cos\theta,\sin\theta,y).$$

Figure 1: The Hudson torus

Let us present a geometric description of this embedding. Define a map $\widetilde a:S^0\times S^{n-1}\to S^{n-1}$$\widetilde a:S^0\times S^{n-1}\to S^{n-1}$ by $\widetilde a(s,t):=s\overline a(t)$$\widetilde a(s,t):=s\overline a(t)$. This map gives an embedding

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$$\displaystyle S^0\times S^{n-1}\xrightarrow{\widetilde a\times {\rm pr}_2}S^{n-1}\times S^{n-1}\subset D^{n+1}\times S^{n-1}\overset{{\rm i}}\to\Rr^{2n}.$$

See Figure 1. The image of

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$\widetilde a\times{\rm pr}_2$ is the union of the graphs of the maps $\overline a$$\overline a$ and $-\overline a$$-\overline a$. For any $t\in S^{n-1}$$t\in S^{n-1}$ the disk

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${\rm i}(D^{n+1}\times t)$ intersects the image at two points lying in

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${\rm i}(D^n\times t)$, i.e., at the image of an embedding

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$S^0\times t\to {\rm i}(D^n\times t)$. The embedding $\Hud_n'(a)$$\Hud_n'(a)$ is obtained by extending the latter embeddings to embeddings

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$S^1\times t\to {\rm i}(D^{n+1}\times t)$ for all $t$$t$. See Figure 2.

Figure 2: To the construction of the Hudson torus

Remark 3.4. (a) The analogue of Proposition 3.2 for $\Hud_n$$\Hud_n$ replaced to $\Hud_n'$$\Hud_n'$ holds, with an analogous proof.

(b) The embeddings $\Hud_n(a)$$\Hud_n(a)$ and $\Hud_n'(a)$$\Hud_n'(a)$ are smoothly isotopic for $n\ge4$$n\ge4$ and are PS isotopic for $n\ge3$$n\ge3$ [Skopenkov2006a, commutativity of the left upper square in the Restriction Lemma 5.2], [Skopenkov2015a, Lemma 2.15.c] (see [Skopenkov2016f, Remark 1.2]). This follows by calculation of the Whitney invariant (Remark 5.3.d below). It would be interesting to know if they are smoothly isotopic for $n=3$$n=3$. It would be interesting to know if they are piecewise smoothly isotopic for $n=2$$n=2$.

(b') Although construction of $\Hud_n(a)$$\Hud_n(a)$ and $\Hud_n'(a)$$\Hud_n'(a)$ is explicit, no explicit construction of an isotopy between them is known. Proof from (b) of their being isotopic is by calculating invariants, and using completeness result for the invariants. This proof is fairly constructive, so in principle it does give a specific isotopy. However, the construction is so complicated that one would not call such an isotopy explicit. This is analogous to the Smale `turning sphere inside out' theorem (although the Smale theorem concerns regular homotopy of immersions not isotopy of embeddings).

In higher dimensions, a proof that any embedding is isotopic to some of the constructed embeddings is even less likely to use an explicit construction. The only proofs we have go through calculating the invariants and using classification results.

(c) For $n=2$$n=2$ Example 3.3 gives what we call the 'left' Hudson torus. The 'right' Hudson torus is constructed analogously. It is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of $S^1\times S^1$$S^1\times S^1$. The right and the left Hudson tori are not isotopic by Remark 5.3.d below.

(d) Analogously one constructs the Hudson torus $\Hud_{n,p}(a):S^p\times S^{n-p}\to\Rr^{2n-p+1}$$\Hud_{n,p}(a):S^p\times S^{n-p}\to\Rr^{2n-p+1}$ for $a\in\Zz$$a\in\Zz$ and $n>p\ge0$$n>p\ge0$ or, more generally, $\Hud_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$$\Hud_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$ for $a\in\pi_n(S^{m-n+p-1})$$a\in\pi_n(S^{m-n+p-1})$ and $m>n>p\ge0$$m>n>p\ge0$. There are versions $\Hud'_{m,n,p}(a)$$\Hud'_{m,n,p}(a)$ of these constructions corresponding to Definition 3.3. For $p=0$$p=0$ this corresponds to the Zeeman map [Skopenkov2016h, Definition 2.2] and its composition with 'the unframed second Kirby move' [Skopenkov2015a, $\S$$\S$2.3]. It would be interesting to know if the links $\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$$\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$ are isotopic, cf. [Skopenkov2015a, Remark 2.7.b]. These constructions can be further generalized [Skopenkov2016k].

Example 3.5. Define a smooth embedding

$\displaystyle l:S^1\times S^{n-1}\to \C^n=\R^{2n}\quad\text{by}\quad l(z,x) := \left(1+\frac z2\right)x.$ $$\displaystyle l:S^1\times S^{n-1}\to \C^n=\R^{2n}\quad\text{by}\quad l(z,x) := \left(1+\frac z2\right)x.$$

Here $S^1\subset\R^2=\C$$S^1\subset\R^2=\C$ and $S^{n-1}\subset\R^n\subset\C^n$$S^{n-1}\subset\R^n\subset\C^n$. This embedding is known to be Lagrangian; see more in [Nemirovski2024].

4 Action by linked embedded connected sum

In this section we generalize the construction of the Hudson torus $\Hud(a)$$\Hud(a)$. Let $N$$N$ be a closed connected oriented $n$$n$-manifold. We work in the smooth category which we omit. Apparently analogous results hold for $n\ge3$$n\ge3$ in the PL and PS categories (see [Skopenkov2016f, Remark 1.2]).

Example 4.1. For any $n\ge4$$n\ge4$, an embedding $f_0:N\to\Rr^{2n}$$f_0:N\to\Rr^{2n}$ and $a\in H_1(N;\Zz)$$a\in H_1(N;\Zz)$, we shall construct an embedding $f_a:N\to\Rr^{2n}$$f_a:N\to\Rr^{2n}$. This embedding is said to be obtained by linked embedded connected sum of $f_0$$f_0$ with an $n$$n$-sphere representing the `homology Alexander dual' $A:=\widehat{A_{f_0}}a\in H_n(C_{f_0})$$A:=\widehat{A_{f_0}}a\in H_n(C_{f_0})$ of $a$$a$ (defined in [Skopenkov2005, Alexander Duality Lemma 4.6]).

Represent $a$$a$ by an embedding $a:S^1\to N$$a:S^1\to N$. By definition, the class $A$$A$ is represented by properly oriented $\nu_{f_0}^{-1}a(S^1)$$\nu_{f_0}^{-1}a(S^1)$. Since any orientable bundle over $S^1$$S^1$ is trivial, $\nu_{f_0}^{-1}a(S^1)\cong S^1\times S^{n-1}$$\nu_{f_0}^{-1}a(S^1)\cong S^1\times S^{n-1}$. Take an embedding $g:S^1\times S^{n-1}\to C_{f_0}$$g:S^1\times S^{n-1}\to C_{f_0}$ whose image is $\nu_{f_0}^{-1}a(S^1)$$\nu_{f_0}^{-1}a(S^1)$ and which represents $A$$A$. By embedded surgery on $S^1\times1_{n-1}\subset S^1\times S^{n-1}$$S^1\times1_{n-1}\subset S^1\times S^{n-1}$ we obtain an embedding $g_1:S^n\to C_{f_0}$$g_1:S^n\to C_{f_0}$ representing $A$$A$ (see details in Proposition 4.2 below). Define $f_a$$f_a$ to be the linked embedded connected sum of $f_0$$f_0$ and $g_1$$g_1$, along some arc joining their images.

Proposition 4.2 (Embedded surgery). For any $n\ge3$$n\ge3$, a neighborhood $U$$U$ of a codimension at least 3 subpolyhedron in $\Rr^{2n}$$\Rr^{2n}$ and an embedding $g:S^1\times S^{n-1}\to\Rr^{2n}-U$$g:S^1\times S^{n-1}\to\Rr^{2n}-U$ there is an embedding $g_1:S^n\to\Rr^{2n}-U$$g_1:S^n\to\Rr^{2n}-U$ homologous to $g$$g$.

Proof. Take a vector field on $g(S^1\times1_{n-1})$$g(S^1\times1_{n-1})$ normal to $g(S^1\times S^{n-1})$$g(S^1\times S^{n-1})$. Extend $g|_{S^1\times1_{n-1}}$$g|_{S^1\times1_{n-1}}$ along this vector field to a map $\overline b:D^2\to\Rr^{2n}$$\overline b:D^2\to\Rr^{2n}$.

Since $2n>4$$2n>4$ and $U$$U$ is a neighborhood $U$$U$ of a codimension at least 3 subpolyhedron, by general position we may assume that $\overline b$$\overline b$ is an embedding and that

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$\overline b({\rm Int}D^2)$ misses $U\cup g(S^1\times S^{n-1})$$U\cup g(S^1\times S^{n-1})$.

Since $n-1>1$$n-1>1$, we have $\pi_1(V_{2n-2,n-1})=0$$\pi_1(V_{2n-2,n-1})=0$. Hence the standard $(n-1)$$(n-1)$-framing of $S^1\times1_{n-1}$$S^1\times1_{n-1}$ in $S^1\times S^{n-1}$$S^1\times S^{n-1}$ extends to an $(n-1)$$(n-1)$-framing on $\overline b(D^2)$$\overline b(D^2)$ in $\Rr^{2n}$$\Rr^{2n}$. Thus $\overline b$$\overline b$ extends to an embedding

$\displaystyle \widehat b:D^2\times D^{n-1}\to\R^{2n}-U\quad\text{such that}\quad \widehat b(\partial D^2\times D^{n-1})\subset g(S^1\times S^{n-1}).$ $$\displaystyle \widehat b:D^2\times D^{n-1}\to\R^{2n}-U\quad\text{such that}\quad \widehat b(\partial D^2\times D^{n-1})\subset g(S^1\times S^{n-1}).$$

Take an embedding $g_1:S^n\to\R^{2n}-U$$g_1:S^n\to\R^{2n}-U$ such that

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$$\displaystyle g_1(S^n)\ =\ g(S^1\times S^{n-1})-\widehat b(\partial D^2\times{\rm Int}D^{n-1}) \bigcup\limits_{\widehat b(\partial D^2\times\partial D^{n-1})} \widehat b(D^2\times\partial D^{n-1})\ \cong\ S^n.$$

with proper orientation so that $g_1$$g_1$ is homologous to $g$$g$. QED

The isotopy class of the embedding $f_a$$f_a$ is independent of the choises in the construction. The independence of the arc and of the maps $g,g_1$$g,g_1$ follows by $n\ge3$$n\ge3$ and by Proposition 4.3 below, respectively.

By Definition 5.1 of the Whitney invariant, $W(f_a,f_0)$$W(f_a,f_0)$ is $a$$a$ for $n\ge3$$n\ge3$ odd and $a \mod2$$a \mod2$ for $n\ge4$$n\ge4$ even. Thus by Theorem 2.1.a for $n\ge4$$n\ge4$ all isotopy classes of embeddings $N\to\Rr^{2n}$$N\to\Rr^{2n}$ can be obtained from any chosen embedding $f_0$$f_0$ by the above construction.

Proposition 4.3. For any $n\ge4$$n\ge4$ both the linked embedded connected sum and parametric connected sum (introduced in [Skopenkov2006a], [Skopenkov2015a]) define free transitive actions of $H_1(N;\Zz_{\varepsilon(n-1)})$$H_1(N;\Zz_{\varepsilon(n-1)})$ on $E^{2n}(N)$$E^{2n}(N)$.

This follows by Theorem 2.1.a and by [Skopenkov2014, Remark 18.a].

5 The Whitney invariant

Let $N$$N$ be a closed $n$$n$-manifold. Take an embedding $f_0:N\to\Rr^m$$f_0:N\to\Rr^m$. Fix an orientation on $\Rr^m$$\Rr^m$. For any other embedding $f \colon N \to \Rr^m$$f \colon N \to \Rr^m$ we define the Whitney invariant

$\displaystyle W(f, f_0)=W_{f_0}(f)=W(f)\in H_{2n-m+1}(N;\Zz_N).$ $$\displaystyle W(f, f_0)=W_{f_0}(f)=W(f)\in H_{2n-m+1}(N;\Zz_N).$$

Here the coefficients $\Zz_N$$\Zz_N$ are $\Zz$$\Zz$ if $N$$N$ is oriented and $m-n$$m-n$ is odd, and are $\Zz_2$$\Zz_2$ otherwise.

Roughly speaking,

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$W(f):=[{\rm Cl}\Sigma(H)]$ is defined as the homology class of the closure of the self-intersection set of a general position homotopy $H$$H$ between $f$$f$ and $f_0$$f_0$.

This is formalized in Definition 5.2 in the smooth category, following [Skopenkov2010], see also [Haefliger&Hirsch1963]. The definition in the PL category is analogous [Hudson1969, $\S$$\S$11], [Vrabec1977, p. 145], [Skopenkov2006, $\S$$\S$2.4 `The Whitney invariant']. We begin by presenting a simpler definition, Definition 5.1, for a particular case.

For Theorem 2.1 only the case $m=2n$$m=2n$ is required.

Definition 5.1. Assume that $N$$N$ is $(2n-m)$$(2n-m)$-connected and $2m\ge3n+3$$2m\ge3n+3$. Then by [Haefliger&Hirsch1963, Theorem 3.1.b] restrictions of $f$$f$ and $f_0$$f_0$ to $N_0$$N_0$ are isotopic, cf. [Takase2006, Lemma 2.2]. (Here is sketch of an argument. Using the Smale-Hirsch classification of immersions we obtain that restrictions of $f$$f$ and $f_0$$f_0$ to $N_0$$N_0$ are `regular homotopic', see [Koschorke2013, Definition 2.7]. Since $N$$N$ is $(2n-m)$$(2n-m)$-connected, $N_0$$N_0$ retracts to an $(m-n-1)$$(m-n-1)$-dimensional polyhedron. Therefore these restrictions are isotopic.)

So we can make an isotopy of $f$$f$ and assume that $f=f_0$$f=f_0$ on $N_0$$N_0$. Take a general position homotopy $F:B^n\times I\to\Rr^m$$F:B^n\times I\to\Rr^m$ relative to $\partial B^n$$\partial B^n$ between the restrictions of $f$$f$ and $f_0$$f_0$ to $B^n$$B^n$. Let $f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$$f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$ (`the intersection of this homotopy with $f(N-B^n)$$f(N-B^n)$').

Since $n+2(n+1)<2m$$n+2(n+1)<2m$, by general position

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${\rm Cl}(f\cap F)$ is a compact $(2n+1-m)$$(2n+1-m)$-manifold whose boundary is contained in $\partial N_0$$\partial N_0$.

So $f\cap F$$f\cap F$ carries a homology class with $\Zz_2$$\Zz_2$ coefficients. If $m-n$$m-n$ is odd and $N$$N$ is oriented, then $f\cap F$$f\cap F$ has a natural orientation defined below, and so carries a homology class with $\Zz$$\Zz$ coefficients. Define $W(f)$$W(f)$ to be the homology class:

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$$\displaystyle W(f):=[{\rm Cl}(f\cap F)]\in H_{2n-m+1}(N_0,\partial N_0;\Zz_N)\cong H_{2n-m+1}(N;\Zz_N).$$

The orientation on $f\cap F$$f\cap F$ (extendable to

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${\rm Cl}(f\cap F)$) is defined (for $m-n$$m-n$ odd and $N$$N$ is oriented) as follows (cf. Remark 7.3). For any point $x_f\in f\cap F$$x_f\in f\cap F$ take a base $\xi_f$$\xi_f$ at $x_f$$x_f$ tangent to $f\cap F$$f\cap F$. Complete this base to a positive base $(\xi_f,\eta_f)$$(\xi_f,\eta_f)$ tangent to $N$$N$. Since $n+2(n+1)<2m$$n+2(n+1)<2m$, by general position there is a unique point $x_F\in B^n\times I$$x_F\in B^n\times I$ such that $Fx_F=fx_f$$Fx_F=fx_f$. The tangent base $\xi_f$$\xi_f$ at $x_f$$x_f$ thus gives a base $\xi_F$$\xi_F$ at $x_F$$x_F$ tangent to $B^n\times I$$B^n\times I$ such that $df(x_f)\xi_f=dF(x_F)\xi_F$$df(x_f)\xi_f=dF(x_F)\xi_F$. Complete this base $\xi_F$$\xi_F$ to a positive base $(\xi_F,\eta_F)$$(\xi_F,\eta_F)$ tangent to $B^n\times I$$B^n\times I$, where the orientation on $B^n$$B^n$ comes from $N$$N$. The union $\zeta=(df(x_f)\xi_f,df(x_f)\eta_f,dF(x_F)\eta_F)$$\zeta=(df(x_f)\xi_f,df(x_f)\eta_f,dF(x_F)\eta_F)$ of the images of the constructed bases is a base at $fx_f=Fx_F$$fx_f=Fx_F$ of $\Rr^m$$\Rr^m$. If $\zeta$$\zeta$ is positive, then call the tangent base $\xi_f$$\xi_f$ of $f\cap F$$f\cap F$ `positive'. Since a change of the orientation on $f\cap F$$f\cap F$ forces a change of the orientation of $\zeta$$\zeta$, this condition indeed defines an orientation on $f\cap F$$f\cap F$.

Definition 5.2. Assume that $m\ge n+2$$m\ge n+2$. Take a general position homotopy $H:N\times I\to\Rr^m\times I$$H:N\times I\to\Rr^m\times I$ between $f_0$$f_0$ and $f$$f$.

The closure

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${\rm Cl}\Sigma(H)$ of the self-intersection set carries a cycle mod 2. If $N$$N$ is oriented and $m-n$$m-n$ is odd, the closure also carries an integer cycle. See [Hudson1967, $\S$$\S$11], [Skopenkov2006, $\S$$\S$2.3 `The Whitney obstruction']. Let us informally explain these facts. For $2m\ge3n+2$$2m\ge3n+2$ by general position the closure

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${\rm Cl}\Sigma(H)$ can be assumed to be a submanifold. In general, since $m\ge n+2$$m\ge n+2$, by general position the closure has codimension 2 singularities, as defined in $\S$$\S$7. So the closure carries a cycle mod 2. When $m-n$$m-n$ is odd the closure also has a canonical orientation (see Definition 7.1 and Remark 7.2), so the closure carries an integer cycle.

Define the Whitney invariant to be the homology class:

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$$\displaystyle W(f):=[{\rm Cl}\Sigma(H)]\in H_{2n-m+1}(N\times I;\Zz_N)\cong H_{2n-m+1}(N;\Zz_N).$$

Clearly, $W(f) = W(f')$$W(f) = W(f')$ if $f$$f$ is isotopic to $f'$$f'$. Hence the Whitney invariant defines a map

$\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_N),\quad [f] \mapsto W(f).$ $$\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_N),\quad [f] \mapsto W(f).$$

Clearly, $W(f_0)=0$$W(f_0)=0$ (for both definitions).

The definition of $W$$W$ depends on the choice of $f_0$$f_0$, but we write $W$$W$ not $W_{f_0}$$W_{f_0}$ for brevity.

Remark 5.3. (a) The Whitney invariant is well-defined by Definition 5.2, i.e. is independent of the choice of a general position homotopy $H:N\times I\to\Rr^m\times I$$H:N\times I\to\Rr^m\times I$ from $f_0$$f_0$ to $f$$f$.

This follows from the equality

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$[{\rm Cl}\Sigma(H_0)]−[{\rm Cl}\Sigma(H_1)] = \partial [{\rm Cl}\Sigma(H_{01})]$ for a general position homotopy $H_{01}:N\times I\times I\to\Rr^m\times I\times I$$H_{01}:N\times I\times I\to\Rr^m\times I\times I$ between general position homotopies $H_0,H_1:N\times I\to\Rr^m\times I$$H_0,H_1:N\times I\to\Rr^m\times I$ from $f_0$$f_0$ to $f$$f$. See details in [Hudson1969, $\S$$\S$11].

(b) Definition 5.1 is a particular case of Definition 5.2. (Indeed, if $f=f_0$$f=f_0$ on $N_0$$N_0$, we can take $H$$H$ to be fixed on $N_0$$N_0$. See details in [Skopenkov2010, Difference Lemma 2.4].) Hence the Whitney invariant is well-defined by Definition 5.1, i.e. independent of the choice of $F$$F$ and of the isotopy making $f=f_0$$f=f_0$ outside $B^n$$B^n$.

(c) The class $W(f)$$W(f)$ is independent of the choice of the orientation on $N$$N$ (because a change of the orientation on $N$$N$ forces a change of the orientation on $f\cap F$$f\cap F$ or on

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${\rm Cl}\Sigma(H)$). For the reflection $\sigma:\Rr^m\to\Rr^m$$\sigma:\Rr^m\to\Rr^m$ with respect to a hyperplane we have $W(\sigma\circ f)=-W(f)$$W(\sigma\circ f)=-W(f)$ (because a change of the orientation on $\Rr^m$$\Rr^m$ forces a change of the orientation on $f\cap F$$f\cap F$ or on

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${\rm Cl}\Sigma(H)$; for Definition 5.1 also observe that we may assume that $f=f_0=\sigma\circ f$$f=f_0=\sigma\circ f$ on $N_0$$N_0$).

(d) For the Hudson tori $W(\Hud_n(a))=W(\Hud'_n(a))$$W(\Hud_n(a))=W(\Hud'_n(a))$ is $a$$a$ or $a\mod2$$a\mod2$ for $n\ge3$$n\ge3$, and $W(\Hud'_2(a))=(a\mod2,0)$$W(\Hud'_2(a))=(a\mod2,0)$.

For $\Hud_n(a)$$\Hud_n(a)$ this is clear by Definition 5.1. For $\Hud'_n(a)$$\Hud'_n(a)$ and $n\ge3$$n\ge3$ this was proved in [Hudson1963] (with a different but equivalent definition of the Whitney invariant; using and proving a particular case of Remark 5.3.f). For $\Hud'_2(a)$$\Hud'_2(a)$ the proof is analogous.

(e) $W(f\#g)=W(f)$$W(f\#g)=W(f)$ for any pair of embeddings $f:N\to\Rr^m$$f:N\to\Rr^m$ and $g:S^n\to\Rr^m$$g:S^n\to\Rr^m$. This is clear by Definition 5.1 because $W(f\#g)-W(f)=W(f\#g,f_0)-W(f,f_0)=W(f\#g,f)=0$$W(f\#g)-W(f)=W(f\#g,f_0)-W(f,f_0)=W(f\#g,f)=0$. Let us prove the latter equality. Take the identical isotopy $H_f$$H_f$ of $f$$f$ and a general position homotopy $H_g$$H_g$ between $g$$g$ and the standard embedding. Then the boundary connected sum $H_f\sharp H_g$$H_f\sharp H_g$ is a general position homotopy between $f\#g$$f\#g$ and an embedding isotopic to $f$$f$. The cycle

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${\rm Cl}\Sigma(H_f\sharp H_g)={\rm Cl}\Sigma H_g$ is null-homologous in $S^n$$S^n$ and hence in $N\cong N\#S^n$$N\cong N\#S^n$; cf. [Skopenkov2008, Addendum to the Classification Theorem].

(f) For $m=2n+1$$m=2n+1$ and $N=S^n\sqcup S^n$$N=S^n\sqcup S^n$ the Whitney invariant equals to the pair of linking coefficients [Skopenkov2016h, $\S$$\S$3].

(g) The Whitney invariant need not be a bijection for $m<2n$$m<2n$. This is seen, for example, by applying Theorem 6.4 below in case of knotted tori [Skopenkov2016k, Theorem 5.1]) or by taking $n$$n$ even, $N$$N$ non-orientable, $m=2n$$m=2n$ and applying by Theorem 2.1.b.

(h) The paper [Haefliger&Hirsch1963, $\S$$\S$2] gives yet another equivalent definition of the Whitney invariant, and proves its injectivity under certain restrictions (the Haefliger-Hirsch $h$$h$-principle, see Theorem 6.2).

(i) Let $N$$N$ be a closed orientable $n$$n$-manifold. For two embeddings $f,g:N\to\R^{2n-1}$$f,g:N\to\R^{2n-1}$ the invariant defined in [Haefliger&Hirsch1963, $\S$$\S$2] equals plus or minus their Whitney invariant $W(f,g)$$W(f,g)$ from Definition 5.1. This is proved analogously to the following version.

Let $D^n$$D^n$ and $B^n$$B^n$ be closed $n$$n$-balls in $N$$N$ such that

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$D^n\subset{\rm Int}B^n$. Let $f:N\to\R^{2n-1}$$f:N\to\R^{2n-1}$ a general position immersion which is an embedding on the closure of $N-D^n$$N-D^n$. Let $N_0 := N-B^n$$N_0 := N-B^n$ and $f_0:=f|_{N-B^n}$$f_0:=f|_{N-B^n}$. We omit $\Z_{(n)}$$\Z_{(n)}$-coefficients. Then $f_0^{-1}f(D^n)$$f_0^{-1}f(D^n)$ represents a 1-cycle in $N$$N$. The homology class of this 1-cycle equals minus the normal Stiefel-Whitney class $PD\overline{W}^{n-1}(N)\in H_1(N)$$PD\overline{W}^{n-1}(N)\in H_1(N)$. This follows because

$\displaystyle 0 \overset{(1)}= [f^{-1}f'(N)] \overset{(2)}= [f_0^{-1}f'(N_0)]+[f_0^{-1}f'(B^n)] \overset{(3)} = PD\overline{W}^{n-1}(N)+[f_0^{-1}f(D^n)].$ $$\displaystyle 0 \overset{(1)}= [f^{-1}f'(N)] \overset{(2)}= [f_0^{-1}f'(N_0)]+[f_0^{-1}f'(B^n)] \overset{(3)} = PD\overline{W}^{n-1}(N)+[f_0^{-1}f(D^n)].$$

Here

• $f':N\to\R^{2n-1}$$f':N\to\R^{2n-1}$ is the map formed by a general position small length vector field normal to $f$$f$, and non-zero over $f(B^n)$$f(B^n)$;

• equality (1) holds because $f'$$f'$ is homotopic to a map whose image is disjoint from $f$$f$;

• equality (2) holds because $f_0$$f_0$-preimages of both $f'(N_0)$$f'(N_0)$ and $f'B^n$$f'B^n$ are 1-cycles, and $f|_{B^n}$$f|_{B^n}$-preimage of $f'(N)$$f'(N)$ is a 1-cycle which is null-homologous;

• equality (3) holds because the Euler class of a bundle is dual to the homology class of zeroes of a general position section, and because $[f_0^{-1}f'(B^n)] = [f_0^{-1}f'(D^n)] \overset{(4)}= [f_0^{-1}f(D^n)]$$[f_0^{-1}f'(B^n)] = [f_0^{-1}f'(D^n)] \overset{(4)}= [f_0^{-1}f(D^n)]$;

• equality (4) holds because $f(D^n)$$f(D^n)$ is homologous relative to the boundary and outside $f(N_0)$$f(N_0)$ to the union of $f'(D^n)$$f'(D^n)$ and the the normal vectors at points of $f(\partial D^n)$$f(\partial D^n)$.

6 A generalization to highly-connected manifolds

In this section let $N$$N$ be a closed orientable homologically $k$$k$-connected $n$$n$-manifold, $k\ge0$$k\ge0$. Recall the unknotting theorem [Skopenkov2016c, Theorem 2.4] that all embeddings $N \to\Rr^m$$N \to\Rr^m$ are isotopic when $m\ge 2n-k+1$$m\ge 2n-k+1$ and $n\ge2k+2$$n\ge2k+2$. In this section we generalize Theorem 2.1 to a description of $E^{2n-k}(N)$$E^{2n-k}(N)$ and further to $E^m(N)$$E^m(N)$ for $m\ge2n-2k+1$$m\ge2n-2k+1$.

6.1 Examples

Some simple examples are the Hudson tori $\Hud_{m,n,p}(a), \Hud'_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$$\Hud_{m,n,p}(a), \Hud'_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$.

Example 6.1 (cf. [Skopenkov2010, Definition 1.4]). Assume that $N$$N$ is $k$$k$-connected and $n-k\ge3$$n-k\ge3$. Then for an embedding $f_0:N\to\Rr^{2n-k}$$f_0:N\to\Rr^{2n-k}$ and a class $a\in H_{k+1}(N;\Zz)$$a\in H_{k+1}(N;\Zz)$ one can construct an embedding $f_a:N\to\Rr^{2n-k}$$f_a:N\to\Rr^{2n-k}$ by linked embedded connected sum analogously to the case $k=0$$k=0$ presented in Example 4.1.

We have $W(f_a,f_0)=a$$W(f_a,f_0)=a$ for the Whitney invariant. Hence by Theorem 6.2 below this construction gives a free transitive action of $H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$$H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$ on $E^{2n-k}(N)$$E^{2n-k}(N)$, provided $n\ge k+3$$n\ge k+3$ or $n\ge2k+4$$n\ge2k+4$ in the PL or smooth categories, respectively.

The embedding $f_a$$f_a$ has an alternative construction using parametric connected sum [Skopenkov2014, Remark 18.a].

6.2 Classification

Theorem 6.2. Let $N$$N$ be a closed oriented homologically $k$$k$-connected $n$$n$-manifold, $k\ge0$$k\ge0$. Then the Whitney invariant

$\displaystyle W:E^{2n-k}(N)\to H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$ $$\displaystyle W:E^{2n-k}(N)\to H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$$

is a bijection, provided $n\ge2k+4$$n\ge2k+4$ in the smooth category or $n\ge k+3$$n\ge k+3$ in the PL category.

This was proved for $k$$k$-connected manifolds in the smooth category [Haefliger&Hirsch1963, Theorem 2.4], and in the PL category in [Weber1967], [Hudson1969, $\S$$\S$11], cf. [Boechat&Haefliger1970, Theorem 1.6], [Boechat1971, Theorem 4.2], [Vrabec1977, Theorems 1.1 and 1.2], [Adachi1993, $\S$$\S$7]. The proof actually used the homological $k$$k$-connectedness assumption (basically because the $k$$k$-connectedness was used to ensure high enough connectedness of the complement in $\Rr^m$$\Rr^m$ to the image of $N$$N$, by Alexander duality and simple connectedness of the complement, so homological $k$$k$-connectedness of $N$$N$ is sufficient).

For $k=0$$k=0$ Theorem 6.2 is covered by Theorem 2.1; for $k\ge1$$k\ge1$ it is not. For $k+3\le n\le2k+1$$k+3\le n\le2k+1$ the PL case of Theorem 6.2 gives nothing but the Zeeman Unknotting Spheres Theorem [Skopenkov2016c, Theorem 2.3]. For the case of knotted tori see [Skopenkov2016k, Theorem 3.1].

An inverse to the map $W$$W$ of Theorem 6.2 is given by Example 6.1.

Because of the existence of knotted spheres the analogues of Theorem 6.2 for $n=k+2$$n=k+2$ in the PL case, and for $n\le2k+3$$n\le2k+3$ in the smooth case are false. So for the smooth category, $n\le2k+3$$n\le2k+3$ and $N$$N$ closed connected, a classification of $E^{2n-k}(N)$$E^{2n-k}(N)$ is much harder: for 40 years the only known complete readily calculable classification results were for homology spheres $N$$N$. E.g. $E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)}$$E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)}$ for any $s\ge2$$s\ge2$ [Haefliger1966, Corollary 8.14], [Skopenkov2016s, Theorem 3.2]. The following result for $n=2k+3$$n=2k+3$ was obtained using the Boéchat-Haefliger formula for the smoothing obstruction [Boechat&Haefliger1970, Theorem 2.1], [Boechat1971, Theorem 5.1]. Using that formula one can define the higher-dimensional Kreck invariant [Skopenkov2008, $\S$$\S$4].

Theorem 6.3 [Skopenkov2008, Higher-dimensional Classification Theorem]. Let $N$$N$ be a closed orientable homologically $(2l-2)$$(2l-2)$-connected $(4l-1)$$(4l-1)$-manifold. Then the Whitney invariant

$\displaystyle W:E^{6l}_D(N)\to H_{2l-1}(N)$ $$\displaystyle W:E^{6l}_D(N)\to H_{2l-1}(N)$$

is surjective and for any $u\in H_{2l-1}(N)$$u\in H_{2l-1}(N)$ the Kreck invariant

$\displaystyle \eta_u:W^{-1}u\to\Zz_{d(u)}$ $$\displaystyle \eta_u:W^{-1}u\to\Zz_{d(u)}$$

is a 1-1 correspondence, where $d(u)$$d(u)$ is the divisibility of the projection of $u$$u$ to the free part of $H_1(N)$$H_1(N)$.

Recall that the divisibility of zero is zero and the divisibility of $x\in G-\{0\}$$x\in G-\{0\}$ is $\max\{d\in\Zz\ | \ \text{there is }x_1\in G: \ x=dx_1\}$$\max\{d\in\Zz\ | \ \text{there is }x_1\in G: \ x=dx_1\}$.

How does one describe $E^m(N)$$E^m(N)$ when $N$$N$ is not $(2n-m)$$(2n-m)$-connected? For general $N$$N$ see the sentence on $E^{2n-1}(N)$$E^{2n-1}(N)$ at the end of $\S$$\S$2. We can say more as the connectivity $k$$k$ of $N$$N$ increases. Some estimations of $E^{2n-k-1}(N)$$E^{2n-k-1}(N)$ for a closed $k$$k$-connected $n$$n$-manifold $N$$N$ are presented in [Skopenkov2010]. For $k>1$$k>1$ one can go even further:

Theorem 6.4 [Becker&Glover1971, Corollary 1.3]. Let $N$$N$ be a closed $k$$k$-connected $n$$n$-manifold embeddable into $\Rr^m$$\Rr^m$, $m\ge2n-2k+1$$m\ge2n-2k+1$ and $2m\ge 3n+4$$2m\ge 3n+4$. Then there is a 1-1 correspondence

$\displaystyle E^m(N)\to [N_0, V_{m,n+1}].$ $$\displaystyle E^m(N)\to [N_0, V_{m,n+1}].$$

The 1-1 correspondence is constructed using `the Haefliger-Wu invariant' involving the configuration space of distinct pairs [Skopenkov2006, $\S$$\S$5]. For $k=0$$k=0$ Theorem 6.4 is the same as General Position Theorem [Skopenkov2016c, Theorem 2.1] (because $V_{2n+1,n+1}$$V_{2n+1,n+1}$ is $(n-1)$$(n-1)$-connected). For $k=1$$k=1$ Theorem 6.4 is covered by Theorem 6.2; for $k\ge2$$k\ge2$ it is not. For application to knotted tori see [Skopenkov2016k, Theorem 5.1]. For generalization to arbitrary manifolds see survey [Skopenkov2006, $\S$$\S$5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Observe that in Theorem 6.4 $V_{m,n+1}$$V_{m,n+1}$ can be replaced by $V_{M,M+n-m+1}$$V_{M,M+n-m+1}$ for any $M>n$$M>n$.

7 An orientation on the self-intersection set

Let $f:N\to\Rr^m$$f:N\to\Rr^m$ be a smooth map from an oriented $n$$n$-manifold $N$$N$ where $m\ge n+2$$m\ge n+2$. We assume that the closure

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${\rm Cl}\Sigma(f)$ of the self-intersection set of $f$$f$ has codimension 2 singularities, i.e., there is

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$P\subset{\rm Cl}\Sigma(f)$ such that

• both $P$$P$ and

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  ${\rm Cl}\Sigma(f)$ are subpolyhedra of some triangulation of $N$$N$,
• we have $\dim P\le\dim\Sigma(f)-2$$\dim P\le\dim\Sigma(f)-2$ and
• $\Sigma(f)-P$$\Sigma(f)-P$ is an open manifold consisting of self-transverse double points of $f$$f$.

Definition 7.1 (A canonical orientation on $\Sigma(f)-P$$\Sigma(f)-P$). Take points $x,y\in N$$x,y\in N$ away from $P$$P$ and such that $fx=fy$$fx=fy$. Then a $(2n-m)$$(2n-m)$-base $\xi_x$$\xi_x$ tangent to $\Sigma(f)-P$$\Sigma(f)-P$ at $x$$x$ gives a $(2n-m)$$(2n-m)$-base $\xi_y:=df_y^{-1}df_x(\xi_x)$$\xi_y:=df_y^{-1}df_x(\xi_x)$ tangent to $\Sigma(f)-P$$\Sigma(f)-P$ at $y$$y$. Since $N$$N$ is oriented, we can take positive $(m-n)$$(m-n)$-bases $\eta_x$$\eta_x$ and $\eta_y$$\eta_y$ at $x$$x$ and $y$$y$ normal to $\xi_x$$\xi_x$ and to $\xi_y$$\xi_y$. If the base $(df_x(\xi_x),df_x(\eta_x),df_y(\eta_y))$$(df_x(\xi_x),df_x(\eta_x),df_y(\eta_y))$ of $\Rr^m$$\Rr^m$ is positive, then call the base $\xi_x$$\xi_x$ positive. This is well-defined because a change of the sign of $\xi_x$$\xi_x$ forces changes of the signs of $\xi_y,\eta_x$$\xi_y,\eta_x$ and $\eta_y$$\eta_y$.

Remark 7.2 (Properties of the orientation just defined on $\Sigma(f) - P$$\Sigma(f) - P$)..

1. A change of the orientation of $N$$N$ forces changes of the signs of $\eta_x$$\eta_x$ and $\eta_y$$\eta_y$ and so does not change the orientation of $\Sigma(f)-P$$\Sigma(f)-P$.
2. The orientation on $\Sigma(f)-P$$\Sigma(f)-P$ need not extend to

   Tex syntax error

   ${\rm Cl}\Sigma(f)$: take the smooth cone $f:D^3\to\Rr^5$$f:D^3\to\Rr^5$ over a general position map $g:S^2\to\Rr^4$$g:S^2\to\Rr^4$ having only two transverse self-intersection points, where the smooth cone is defined by $f(tx):=(g(x)\sin(\pi t/2),\cos(\pi t/2))$$f(tx):=(g(x)\sin(\pi t/2),\cos(\pi t/2))$, for $x\in S^2$$x\in S^2$ and $t\in[0,1]$$t\in[0,1]$.
3. The orientation on $\Sigma(f)-P$$\Sigma(f)-P$ extends to

   Tex syntax error

   ${\rm Cl}\Sigma(f)$ if $m-n$$m-n$ is odd [Hudson1969, Lemma 11.4].

Remark 7.3 (A canonical orientation on $f\Sigma(f)-fP$$f\Sigma(f)-fP$ for $m-n$$m-n$ even). This remark is added as a complement for Definition 7.1 but is not needed for the definition of the Whitney invariant.

Take a $(2n-m)$$(2n-m)$-base $\xi$$\xi$ at a point $x\in f\Sigma(f)-fP$$x\in f\Sigma(f)-fP$. Since $N$$N$ is oriented, we can take a positive $(m-n)$$(m-n)$-base $\eta_+$$\eta_+$ normal to $f\Sigma(f)$$f\Sigma(f)$ in one sheet of $fN$$fN$. Analogously construct an $(m-n)$$(m-n)$-base $\eta_-$$\eta_-$ for the other sheet of $fN$$fN$. Since $m-n$$m-n$ is even, the orientation of the base $(\xi,\eta_+,\eta_-)$$(\xi,\eta_+,\eta_-)$ of $\Rr^m$$\Rr^m$ does not depend on choosing the first and the other sheet of $fN$$fN$ at $x$$x$. If the base $(\xi,\eta_+,\eta_-)$$(\xi,\eta_+,\eta_-)$ is positive, then call the base $\xi$$\xi$ positive. This is well-defined because a change of the sign of $\xi$$\xi$ forces changes of the signs of $\eta_+,\eta_-$$\eta_+,\eta_-$ and so of $(\xi,\eta_+,\eta_-)$$(\xi,\eta_+,\eta_-)$.

We remark that a change of the orientation of $N$$N$ forces changes of the signs of $\eta_+,\eta_-$$\eta_+,\eta_-$ and so does not change the orientation of $f\Sigma(f)-fP$$f\Sigma(f)-fP$.

8 References

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\overset{(1)}= [f^{-1}f'(N)] \overset{(2)}= [f_0^{-1}f'(N_0)]+[f_0^{-1}f'(B^n)] \overset{(3)} = PD\overline{W}^{n-1}(N)+[f_0^{-1}f(D^n)].$$ Here * $f':N\to\R^{2n-1}$ is the map formed by a general position small length vector field normal to $f$, and non-zero over $f(B^n)$; * equality (1) holds because $f'$ is homotopic to a map whose image is disjoint from $f$; * equality (2) holds because $f_0$-preimages of both $f'(N_0)$ and $f'B^n$ are 1-cycles, and $f|_{B^n}$-preimage of $f'(N)$ is a 1-cycle which is null-homologous; * equality (3) holds because the Euler class of a bundle is dual to the homology class of zeroes of a general position section, and because $[f_0^{-1}f'(B^n)] = [f_0^{-1}f'(D^n)] \overset{(4)}= [f_0^{-1}f(D^n)]$; * equality (4) holds because $f(D^n)$ is homologous relative to the boundary and outside $f(N_0)$ to the union of $f'(D^n)$ and the the normal vectors at points of $f(\partial D^n)$. {{endthm}} == A generalization to highly-connected manifolds == ; In this section let $N$ be a closed orientable homologically $k$-connected $n$-manifold, $k\ge0$. Recall the [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting theorems|unknotting theorem]] \cite[Theorem 2.4]{Skopenkov2016c} that all embeddings $N \to\Rr^m$ are isotopic when $m\ge 2n-k+1$ and $n\ge2k+2$. In this section we generalize Theorem \ref{th4} to a description of $E^{2n-k}(N)$ and further to $E^m(N)$ for $m\ge2n-2k+1$. ===Examples=== ; Some simple examples are [[#Hudson_tori|the Hudson tori]] $\Hud_{m,n,p}(a), \Hud'_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$. {{beginthm|Example|(cf. \cite[Definition 1.4]{Skopenkov2010})}}\label{hicoe} Assume that $N$ is $k$-connected and $n-k\ge3$. Then for an embedding $f_0:N\to\Rr^{2n-k}$ and a class $a\in H_{k+1}(N;\Zz)$ one can construct an embedding $f_a:N\to\Rr^{2n-k}$ by linked embedded connected sum analogously to [[#Action_by_linked_embedded_connected_sum|the case $k=0$]] presented in Example \ref{d:lecs}. {{endthm}} We have $W(f_a,f_0)=a$ for [[#The Whitney invariant|the Whitney invariant]]. Hence by Theorem \ref{hico} below this construction gives a free transitive action of $H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$ on $E^{2n-k}(N)$, provided $n\ge k+3$ or $n\ge2k+4$ in the PL or smooth categories, respectively. The embedding $f_a$ has an alternative construction using [[Parametric_connected_sum#Applications|parametric connected sum]] \cite[Remark 18.a]{Skopenkov2014}. === Classification === ; \label{s:just} {{beginthm|Theorem}}\label{hico} Let $N$ be a closed oriented homologically $k$-connected $n$-manifold, $k\ge0$. Then [[#The Whitney invariant|the Whitney invariant]] $$W:E^{2n-k}(N)\to H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$$ is a bijection, provided $n\ge2k+4$ in the smooth category or $n\ge k+3$ in the PL category. {{endthm}} This was proved for $k$-connected manifolds in the smooth category \cite[Theorem 2.4]{Haefliger&Hirsch1963}, and in the PL category in \cite{Weber1967}, \cite[$\S]{Hudson1969}, cf. \cite[Theorem 1.6]{Boechat&Haefliger1970}, \cite[Theorem 4.2]{Boechat1971}, \cite[Theorems 1.1 and 1.2]{Vrabec1977}, \cite[$\S]{Adachi1993}. The proof actually used the ''homological'' $k$-connectedness assumption (basically because the $k$-connectedness was used to ensure high enough connectedness of the complement in $\Rr^m$ to the image of $N$, by Alexander duality and simple connectedness of the complement, so homological $k$-connectedness of $N$ is sufficient). For $k=0$ Theorem \ref{hico} is covered by Theorem \ref{th4}; for $k\ge1$ it is not. For $k+3\le n\le2k+1$ the PL case of Theorem \ref{hico} gives nothing but the Zeeman [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting theorems|Unknotting Spheres Theorem]] \cite[Theorem 2.3]{Skopenkov2016c}. For the case of [[Knotted_tori#Classification_just_below_the_stable_range|knotted tori]] see \cite[Theorem 3.1]{Skopenkov2016k}. An inverse to the map $W$ of Theorem \ref{hico} is given by Example \ref{hicoe}. Because of the existence of knotted spheres the analogues of Theorem \ref{hico} for $n=k+2$ in the PL case, and for $n\le2k+3$ in the smooth case are false. So for the smooth category, $n\le2k+3$ and $N$ closed connected, a classification of $E^{2n-k}(N)$ is much harder: for 40 years the ''only'' known complete readily calculable classification results were for homology spheres $N$. E.g. $E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)}$ for any $s\ge2$ \cite[Corollary 8.14]{Haefliger1966}, \cite[Theorem 3.2]{Skopenkov2016s}. The following result for $n=2k+3$ was obtained using the Boéchat-Haefliger formula for the smoothing obstruction \cite[Theorem 2.1]{Boechat&Haefliger1970}, \cite[Theorem 5.1]{Boechat1971}. Using that formula one can define the higher-dimensional [[3-manifolds in 6-space#The Kreck invariant|Kreck invariant]] \cite[$\S]{Skopenkov2008}. {{beginthm|Theorem|\cite[Higher-dimensional Classification Theorem]{Skopenkov2008}}}\label{hicod} Let $N$ be a closed orientable homologically $(2l-2)$-connected $(4l-1)$-manifold. Then [[#The Whitney invariant|the Whitney invariant]] $$ W:E^{6l}_D(N)\to H_{2l-1}(N) $$ is surjective and for any $u\in H_{2l-1}(N)$ the Kreck invariant $$ \eta_u:W^{-1}u\to\Zz_{d(u)} $$ is a 1-1 correspondence, where $d(u)$ is the divisibility of the projection of $u$ to the free part of $H_1(N)$. {{endthm}} Recall that the divisibility of zero is zero and the divisibility of $x\in G-\{0\}$ is $\max\{d\in\Zz\ | \ \text{there is }x_1\in G: \ x=dx_1\}$. How does one describe $E^m(N)$ when $N$ is not $(2n-m)$-connected? For general $N$ see the [[#Classification|sentence]] on $E^{2n-1}(N)$ at the end of $\S$\ref{s:e-cla}. We can say more as the connectivity $k$ of $N$ increases. Some estimations of $E^{2n-k-1}(N)$ for a closed $k$-connected $n$-manifold $N$ are presented in \cite{Skopenkov2010}. For $k>1$ one can go even further: {{beginthm|Theorem|\cite[Corollary 1.3]{Becker&Glover1971}}}\label{begl} Let $N$ be a closed $k$-connected $n$-manifold embeddable into $\Rr^m$, $m\ge2n-2k+1$ and m\ge 3n+4$. Then there is a 1-1 correspondence $$E^m(N)\to [N_0, V_{m,n+1}].$$ {{endthm}} The 1-1 correspondence is constructed using `the Haefliger-Wu invariant' involving the configuration space of distinct pairs \cite[$\S]{Skopenkov2006}. For $k=0$ Theorem \ref{begl} is the same as [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting_theorems|General Position Theorem]] \cite[Theorem 2.1]{Skopenkov2016c} (because $V_{2n+1,n+1}$ is $(n-1)$-connected). For $k=1$ Theorem \ref{begl} is covered by Theorem \ref{hico}; for $k\ge2$ it is not. For application to [[Knotted_tori#Further_classification|knotted tori]] see \cite[Theorem 5.1]{Skopenkov2016k}. For generalization to arbitrary manifolds see survey \cite[$\S]{Skopenkov2006} and \cite{Haefliger1963}, \cite{Weber1967}, \cite{Skopenkov2002}. Observe that in Theorem \ref{begl} $V_{m,n+1}$ can be replaced by $V_{M,M+n-m+1}$ for any $M>n$. == An orientation on the self-intersection set == ; \label{s:orien} Let $f:N\to\Rr^m$ be a smooth map from an oriented $n$-manifold $N$ where $m\ge n+2$. We assume that the closure ${\rm Cl}\Sigma(f)$ of the self-intersection set of $f$ ''has codimension 2 singularities'', i.e., there is $P\subset{\rm Cl}\Sigma(f)$ such that * both $P$ and ${\rm Cl}\Sigma(f)$ are subpolyhedra of some triangulation of $N$, * we have $\dim P\le\dim\Sigma(f)-2$ and * $\Sigma(f)-P$ is an open manifold consisting of self-transverse double points of $f$. {{beginthm|Definition|(A canonical orientation on $\Sigma(f)-P$)}}\label{oris} Take points $x,y\in N$ away from $P$ and such that $fx=fy$. Then a $(2n-m)$-base $\xi_x$ tangent to $\Sigma(f)-P$ at $x$ gives a $(2n-m)$-base $\xi_y:=df_y^{-1}df_x(\xi_x)$ tangent to $\Sigma(f)-P$ at $y$. Since $N$ is oriented, we can take positive $(m-n)$-bases $\eta_x$ and $\eta_y$ at $x$ and $y$ normal to $\xi_x$ and to $\xi_y$. If the base $(df_x(\xi_x),df_x(\eta_x),df_y(\eta_y))$ of $\Rr^m$ is positive, then call the base $\xi_x$ positive. This is well-defined because a change of the sign of $\xi_x$ forces changes of the signs of $\xi_y,\eta_x$ and $\eta_y$. {{endthm}} {{beginthm|Remark|(Properties of the orientation just defined on $\Sigma(f) - P$).}}\label{remoris} # A change of the orientation of $N$ forces changes of the signs of $\eta_x$ and $\eta_y$ and so does not change the orientation of $\Sigma(f)-P$. # The orientation on $\Sigma(f)-P$ need not extend to ${\rm Cl}\Sigma(f)$: take the smooth cone $f:D^3\to\Rr^5$ over a general position map $g:S^2\to\Rr^4$ having only two transverse self-intersection points, where the smooth cone is defined by $f(tx):=(g(x)\sin(\pi t/2),\cos(\pi t/2))$, for $x\in S^2$ and $t\in[0,1]$. # The orientation on $\Sigma(f)-P$ extends to ${\rm Cl}\Sigma(f)$ if $m-n$ is odd \cite[Lemma 11.4]{Hudson1969}. {{endthm}} {{beginthm|Remark|(A canonical orientation on $f\Sigma(f)-fP$ for $m-n$ even)}}\label{orifs} This remark is added as a complement for Definition \ref{oris} but is not needed for the definition of the Whitney invariant. Take a $(2n-m)$-base $\xi$ at a point $x\in f\Sigma(f)-fP$. Since $N$ is oriented, we can take a positive $(m-n)$-base $\eta_+$ normal to $f\Sigma(f)$ in one sheet of $fN$. Analogously construct an $(m-n)$-base $\eta_-$ for the other sheet of $fN$. Since $m-n$ is even, the orientation of the base $(\xi,\eta_+,\eta_-)$ of $\Rr^m$ does not depend on choosing the first and the other sheet of $fN$ at $x$. If the base $(\xi,\eta_+,\eta_-)$ is positive, then call the base $\xi$ positive. This is well-defined because a change of the sign of $\xi$ forces changes of the signs of $\eta_+,\eta_-$ and so of $(\xi,\eta_+,\eta_-)$. We remark that a change of the orientation of $N$ forces changes of the signs of $\eta_+,\eta_-$ and so does not change the orientation of $f\Sigma(f)-fP$. {{endthm}} == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]]N is a connected manifold of dimension $n>1$$n>1$, and $m \ge2n+1$$m \ge2n+1$, then every two embeddings $N \to\Rr^m$$N \to\Rr^m$ are isotopic [Skopenkov2016c, Theorem 3.2], [Skopenkov2006, Theorem 2.5]. In this page we summarize the situation for $m=2n\ge6$$m=2n\ge6$ and $N$$N$ is a connected, as well as in some more general situations. For the classification of embeddings of some disconnected manifolds see [Skopenkov2016h].

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$$\S$1, $\S$$\S$3]. Denote $1_n:=(1,0,\ldots,0)\in S^n$$1_n:=(1,0,\ldots,0)\in S^n$.

2 Classification

For the next theorem, the Whitney invariant $W$$W$ is defined in $\S$$\S$5 below.

Theorem 2.1. Assume that $N$$N$ is a closed connected $n$$n$-manifold, and either $n\ge4$$n\ge4$ or $n=3$$n=3$ and we are in the PL category.

(a) If $N$$N$ is oriented, the Whitney invariant,

$\displaystyle W:E^{2n}(N)\to H_1(N;\Zz_{\varepsilon(n-1)}),$ $$\displaystyle W:E^{2n}(N)\to H_1(N;\Zz_{\varepsilon(n-1)}),$$

is a 1-1 correspondence.

(b) If $N$$N$ is non-orientable, then there is a 1-1 correspondence

$\displaystyle E^{2n}(N)\to \begin{cases} H_1(N;\Zz_2) & n\text{ is odd}\\ \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s & n\text{ is even}\end{cases}.$ $$\displaystyle E^{2n}(N)\to \begin{cases} H_1(N;\Zz_2) & n\text{ is odd}\\ \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s & n\text{ is even}\end{cases}.$$

Remark 2.2 (Comments on the proof). Part (a) is proved in [Haefliger&Hirsch1963, Theorem 2.4] in the smooth category, and in [Weber1967, Theorem 4' in $\S$$\S$2], [Hudson1969, $\S$$\S$11], [Vrabec1977, Theorems 1.1 and 1.2] in the PL category, see also [Haefliger1962b, 1.3.e], [Haefliger1963], [Bausum1975, Theorem 43].

Part (b) is proved in [Bausum1975, Theorem 43] in the smooth category. By [Weber1967, Theorems 1 and 1' in $\S$$\S$2], [Skopenkov1997, Theorem 1.1.c] the proof works also in the PL category.

In Part (b) we replaced the kernel $\ker Sq^1$$\ker Sq^1$ from [Bausum1975, Theorem 43] by $\Zz_2^{s-1}$$\Zz_2^{s-1}$. This is possible because, as a specialist could see, $Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$$Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$ is given by multiplication with the first Stefel-Whitney class $w_1(N)$$w_1(N)$ (which equals to the first Wu class $v_1(N)$$v_1(N)$ [Milnor&Stasheff1974, Theorem 11.4]). Since $N$$N$ is non-orientable, $w_1(N)\neq 0$$w_1(N)\neq 0$. So by Poincaré duality, $\ker Sq^1 \cong \Zz_2^{s-1}$$\ker Sq^1 \cong \Zz_2^{s-1}$.

The 1-1 correspondence from (b) can presumably be defined as a (generalized) Whitney invariant, see [Vrabec1977], but the proof used 'the Haefliger-Wu invariant' whose definition can be found e.g. in [Skopenkov2006, $\S$$\S$5]. It would be interesting to check if part (b) is equivalent to different forms of description of $E^{2n}(N)$$E^{2n}(N)$ [Haefliger1962b, 1.3.e], [Haefliger1963], [Vrabec1977, Theorem 1.1].

The classification of smooth embeddings of 3-manifolds in $\Rr^6$$\Rr^6$ is more complicated, see Theorem 6.3 below for $l=1$$l=1$ or [Skopenkov2016t].

Concerning embeddings of connected $n$$n$-manifolds in $\Rr^{2n-1}$$\Rr^{2n-1}$ see [Yasui1984] for $n\ge5$$n\ge5$, [Skopenkov2016f] for $n=4$$n=4$, and [Saeki1999], [Skopenkov2010], [Tonkonog2010] for manifolds with boundary.

Theorem 2.1 is generalized to a description of $E^{2n-k}(N)$$E^{2n-k}(N)$ for closed $k$$k$-connected $n$$n$-manifolds $N$$N$, see Theorem 6.2.

3 Hudson and Lagrangian tori

Together with the Haefliger knotted sphere [Skopenkov2016t, Example 2.1], [Skopenkov2006, Example 3.4], the examples of Hudson tori presented below were the first examples of non-isotopic embeddings in codimension greater than 2. (Hudson's construction [Hudson1963] was not as explicit as those below.) Abbreviate

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${\rm i}_{2n,n-1}$ to just

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${\rm i}$.

Example 3.1. Let us construct, for any $a\in\Zz$$a\in\Zz$ and $n\ge2$$n\ge2$, a smooth embedding

$\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.$ $$\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.$$

We start with the cases $a=0,1$$a=0,1$.

Take the standard inclusion $\partial D^2\subset\partial D^{n+1}$$\partial D^2\subset\partial D^{n+1}$. The 'standard embedding' $\Hud_n(0)$$\Hud_n(0)$ is given by the standard inclusions

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$$\displaystyle \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\overset{{\rm i}}\to\Rr^{2n}.$$

Define the `standard embedding'

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$2{\rm i}:2D^{n+1}\times S^{n-1}\to\Rr^{2n}$ analogously to

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${\rm i}$, where $2$$2$ means homothety with coefficient 2.

Take the embedding $g_1$$g_1$ given by

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$$\displaystyle S^n=2\partial D^{n+1}\times1_{n-1}\subset 2D^{n+1}\times S^{n-1}\overset{2{\rm i}}\to\Rr^{2n}.$$

The segment

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$2{\rm i}([1,2]1_n\times1_{n-1})$ joins the images of $\Hud_n(0)$$\Hud_n(0)$ and $g_1$$g_1$; the interior of this segment misses the images. Let $\Hud_n(1)$$\Hud_n(1)$ be the linked embedded connected sum of $\Hud_n(0)$$\Hud_n(0)$ and $g_1$$g_1$ along this segment, compatible with the orientation, cf. [Avvakumov2017, $\S$$\S$1.5]. (Here 'linked' means that the images of the embeddings are not contained in disjoint cubes, unlike for the unlinked embedded connected sum [Skopenkov2016c, $\S$$\S$5].)

For $a\in\Zz$$a\in\Zz$ we repeat the above construction of $g_1$$g_1$ replacing $2\partial D^{n+1}\times1_{n-1}$$2\partial D^{n+1}\times1_{n-1}$ by $|a|$$|a|$ copies $(1+\frac1k)\partial D^{n+1}\times1_{n-1}$$(1+\frac1k)\partial D^{n+1}\times1_{n-1}$ of $S^n$$S^n$, $k=1,\ldots,|a|$$k=1,\ldots,|a|$. The copies are outside $D^{n+1}\times S^{n-1}$$D^{n+1}\times S^{n-1}$ and are `parallel' to $\partial D^{n+1}\times1_{n-1}$$\partial D^{n+1}\times1_{n-1}$. The copies have the standard orientation for $a>0$$a>0$ or the opposite orientation for $a<0$$a<0$. Then we make embedded connected sum along natural segments joining every $k$$k$-th copy to the $(k+1)$$(k+1)$-th copy. We obtain an embedding $g_a:S^n\to\Rr^{2n}$$g_a:S^n\to\Rr^{2n}$ which has disjoint images with $\Hud_n(0)$$\Hud_n(0)$. Let $\Hud_n(a)$$\Hud_n(a)$ be the linked embedded connected sum of $\Hud_n(0)$$\Hud_n(0)$ and $g_a$$g_a$.

The original motivation for Hudson was that $\Hud_n(1)$$\Hud_n(1)$ is not isotopic to $\Hud_n(0)$$\Hud_n(0)$ for any $n\ge3$$n\ge3$ (this is a particular case of Proposition 3.2 below). One might guess that $\Hud_n(a)$$\Hud_n(a)$ is not isotopic to $\Hud_n(a')$$\Hud_n(a')$ for $a\ne a'$$a\ne a'$ and that a $\Zz$$\Zz$-valued invariant of $E^{2n}(S^1 \times S^{n-1})$$E^{2n}(S^1 \times S^{n-1})$ can be defined by the homotopy class of the map

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$$\displaystyle S^n\overset g\to S^{2n}-{\rm i}(D^{n+1}\times S^{n-1})\sim S^{2n}-S^{n-1}\sim S^n \quad\text{which is}\quad a\in\pi_n(S^n)\cong\Zz.$$

However, this is only true for $n$$n$ odd.

Proposition 3.2. For $n\ge3$$n\ge3$ odd $\Hud_n(a)$$\Hud_n(a)$ is isotopic to $\Hud_n(a')$$\Hud_n(a')$ if and only if $a=a'$$a=a'$.

For $n\ge4$$n\ge4$ even $\Hud_n(a)$$\Hud_n(a)$ is isotopic to $\Hud_n(a')$$\Hud_n(a')$ if and only if $a\equiv a'\text{ mod}2$$a\equiv a'\text{ mod}2$.

Proposition 3.2 follows by calculation of the Whitney invariant (Remark 5.3.d below) and, for $n$$n$ even, by Theorem 2.1. This proposition holds with the same proof in the piecewise smooth category, whose definition is recalled in [Skopenkov2016f, Remark 1.1]). Proposition 3.2 also holds in the PL category (with an analogous construction of $\Hud_n(a)$$\Hud_n(a)$ for the PL category). It would be interesting to find an explicit construction of an isotopy between $\Hud_{2k}(a)$$\Hud_{2k}(a)$ and $\Hud_{2k}(a+2)$$\Hud_{2k}(a+2)$, cf. [Vrabec1977, $\S$$\S$5]. Analogously, $\Hud_2(a)$$\Hud_2(a)$ is not isotopic to $\Hud_2(a')$$\Hud_2(a')$ if $a\not\equiv a'\text{ mod}2$$a\not\equiv a'\text{ mod}2$. It would be interesting to know if the converse holds, e.g. is $\Hud_2(0)$$\Hud_2(0)$ (PS or smoothly) isotopic to $\Hud_2(2)$$\Hud_2(2)$.

Example 3.3. Take any $a\in\Zz$$a\in\Zz$. Take a map $\overline a:S^{n-1}\to S^{n-1}$$\overline a:S^{n-1}\to S^{n-1}$ of degree $a$$a$ (so we can take $\overline 1=\id$$\overline 1=\id$). Recall that $D^{n+1}=\{(y,x)\in D^n\times D^1\ :\ |y|^2+|x|^2\le1\}$$D^{n+1}=\{(y,x)\in D^n\times D^1\ :\ |y|^2+|x|^2\le1\}$. Define the smooth embedding $\Hud_n'(a)$$\Hud_n'(a)$ to be the composition

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$$\displaystyle S^1\times S^{n-1}\overset{\widehat a}\to D^{n+1}\times S^{n-1}\overset{\rm i}\to\Rr^{2n} \quad\text{where}\quad \widehat a(e^{i\theta},y):=(\overline a(y)\cos\theta,\sin\theta,y).$$

Figure 1: The Hudson torus

Let us present a geometric description of this embedding. Define a map $\widetilde a:S^0\times S^{n-1}\to S^{n-1}$$\widetilde a:S^0\times S^{n-1}\to S^{n-1}$ by $\widetilde a(s,t):=s\overline a(t)$$\widetilde a(s,t):=s\overline a(t)$. This map gives an embedding

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$$\displaystyle S^0\times S^{n-1}\xrightarrow{\widetilde a\times {\rm pr}_2}S^{n-1}\times S^{n-1}\subset D^{n+1}\times S^{n-1}\overset{{\rm i}}\to\Rr^{2n}.$$

See Figure 1. The image of

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$\widetilde a\times{\rm pr}_2$ is the union of the graphs of the maps $\overline a$$\overline a$ and $-\overline a$$-\overline a$. For any $t\in S^{n-1}$$t\in S^{n-1}$ the disk

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${\rm i}(D^{n+1}\times t)$ intersects the image at two points lying in

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${\rm i}(D^n\times t)$, i.e., at the image of an embedding

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$S^0\times t\to {\rm i}(D^n\times t)$. The embedding $\Hud_n'(a)$$\Hud_n'(a)$ is obtained by extending the latter embeddings to embeddings

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$S^1\times t\to {\rm i}(D^{n+1}\times t)$ for all $t$$t$. See Figure 2.

Figure 2: To the construction of the Hudson torus

Remark 3.4. (a) The analogue of Proposition 3.2 for $\Hud_n$$\Hud_n$ replaced to $\Hud_n'$$\Hud_n'$ holds, with an analogous proof.

(b) The embeddings $\Hud_n(a)$$\Hud_n(a)$ and $\Hud_n'(a)$$\Hud_n'(a)$ are smoothly isotopic for $n\ge4$$n\ge4$ and are PS isotopic for $n\ge3$$n\ge3$ [Skopenkov2006a, commutativity of the left upper square in the Restriction Lemma 5.2], [Skopenkov2015a, Lemma 2.15.c] (see [Skopenkov2016f, Remark 1.2]). This follows by calculation of the Whitney invariant (Remark 5.3.d below). It would be interesting to know if they are smoothly isotopic for $n=3$$n=3$. It would be interesting to know if they are piecewise smoothly isotopic for $n=2$$n=2$.

(b') Although construction of $\Hud_n(a)$$\Hud_n(a)$ and $\Hud_n'(a)$$\Hud_n'(a)$ is explicit, no explicit construction of an isotopy between them is known. Proof from (b) of their being isotopic is by calculating invariants, and using completeness result for the invariants. This proof is fairly constructive, so in principle it does give a specific isotopy. However, the construction is so complicated that one would not call such an isotopy explicit. This is analogous to the Smale `turning sphere inside out' theorem (although the Smale theorem concerns regular homotopy of immersions not isotopy of embeddings).

In higher dimensions, a proof that any embedding is isotopic to some of the constructed embeddings is even less likely to use an explicit construction. The only proofs we have go through calculating the invariants and using classification results.

(c) For $n=2$$n=2$ Example 3.3 gives what we call the 'left' Hudson torus. The 'right' Hudson torus is constructed analogously. It is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of $S^1\times S^1$$S^1\times S^1$. The right and the left Hudson tori are not isotopic by Remark 5.3.d below.

(d) Analogously one constructs the Hudson torus $\Hud_{n,p}(a):S^p\times S^{n-p}\to\Rr^{2n-p+1}$$\Hud_{n,p}(a):S^p\times S^{n-p}\to\Rr^{2n-p+1}$ for $a\in\Zz$$a\in\Zz$ and $n>p\ge0$$n>p\ge0$ or, more generally, $\Hud_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$$\Hud_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$ for $a\in\pi_n(S^{m-n+p-1})$$a\in\pi_n(S^{m-n+p-1})$ and $m>n>p\ge0$$m>n>p\ge0$. There are versions $\Hud'_{m,n,p}(a)$$\Hud'_{m,n,p}(a)$ of these constructions corresponding to Definition 3.3. For $p=0$$p=0$ this corresponds to the Zeeman map [Skopenkov2016h, Definition 2.2] and its composition with 'the unframed second Kirby move' [Skopenkov2015a, $\S$$\S$2.3]. It would be interesting to know if the links $\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$$\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$ are isotopic, cf. [Skopenkov2015a, Remark 2.7.b]. These constructions can be further generalized [Skopenkov2016k].

Example 3.5. Define a smooth embedding

$\displaystyle l:S^1\times S^{n-1}\to \C^n=\R^{2n}\quad\text{by}\quad l(z,x) := \left(1+\frac z2\right)x.$ $$\displaystyle l:S^1\times S^{n-1}\to \C^n=\R^{2n}\quad\text{by}\quad l(z,x) := \left(1+\frac z2\right)x.$$

Here $S^1\subset\R^2=\C$$S^1\subset\R^2=\C$ and $S^{n-1}\subset\R^n\subset\C^n$$S^{n-1}\subset\R^n\subset\C^n$. This embedding is known to be Lagrangian; see more in [Nemirovski2024].

4 Action by linked embedded connected sum

In this section we generalize the construction of the Hudson torus $\Hud(a)$$\Hud(a)$. Let $N$$N$ be a closed connected oriented $n$$n$-manifold. We work in the smooth category which we omit. Apparently analogous results hold for $n\ge3$$n\ge3$ in the PL and PS categories (see [Skopenkov2016f, Remark 1.2]).

Example 4.1. For any $n\ge4$$n\ge4$, an embedding $f_0:N\to\Rr^{2n}$$f_0:N\to\Rr^{2n}$ and $a\in H_1(N;\Zz)$$a\in H_1(N;\Zz)$, we shall construct an embedding $f_a:N\to\Rr^{2n}$$f_a:N\to\Rr^{2n}$. This embedding is said to be obtained by linked embedded connected sum of $f_0$$f_0$ with an $n$$n$-sphere representing the `homology Alexander dual' $A:=\widehat{A_{f_0}}a\in H_n(C_{f_0})$$A:=\widehat{A_{f_0}}a\in H_n(C_{f_0})$ of $a$$a$ (defined in [Skopenkov2005, Alexander Duality Lemma 4.6]).

Represent $a$$a$ by an embedding $a:S^1\to N$$a:S^1\to N$. By definition, the class $A$$A$ is represented by properly oriented $\nu_{f_0}^{-1}a(S^1)$$\nu_{f_0}^{-1}a(S^1)$. Since any orientable bundle over $S^1$$S^1$ is trivial, $\nu_{f_0}^{-1}a(S^1)\cong S^1\times S^{n-1}$$\nu_{f_0}^{-1}a(S^1)\cong S^1\times S^{n-1}$. Take an embedding $g:S^1\times S^{n-1}\to C_{f_0}$$g:S^1\times S^{n-1}\to C_{f_0}$ whose image is $\nu_{f_0}^{-1}a(S^1)$$\nu_{f_0}^{-1}a(S^1)$ and which represents $A$$A$. By embedded surgery on $S^1\times1_{n-1}\subset S^1\times S^{n-1}$$S^1\times1_{n-1}\subset S^1\times S^{n-1}$ we obtain an embedding $g_1:S^n\to C_{f_0}$$g_1:S^n\to C_{f_0}$ representing $A$$A$ (see details in Proposition 4.2 below). Define $f_a$$f_a$ to be the linked embedded connected sum of $f_0$$f_0$ and $g_1$$g_1$, along some arc joining their images.

Proposition 4.2 (Embedded surgery). For any $n\ge3$$n\ge3$, a neighborhood $U$$U$ of a codimension at least 3 subpolyhedron in $\Rr^{2n}$$\Rr^{2n}$ and an embedding $g:S^1\times S^{n-1}\to\Rr^{2n}-U$$g:S^1\times S^{n-1}\to\Rr^{2n}-U$ there is an embedding $g_1:S^n\to\Rr^{2n}-U$$g_1:S^n\to\Rr^{2n}-U$ homologous to $g$$g$.

Proof. Take a vector field on $g(S^1\times1_{n-1})$$g(S^1\times1_{n-1})$ normal to $g(S^1\times S^{n-1})$$g(S^1\times S^{n-1})$. Extend $g|_{S^1\times1_{n-1}}$$g|_{S^1\times1_{n-1}}$ along this vector field to a map $\overline b:D^2\to\Rr^{2n}$$\overline b:D^2\to\Rr^{2n}$.

Since $2n>4$$2n>4$ and $U$$U$ is a neighborhood $U$$U$ of a codimension at least 3 subpolyhedron, by general position we may assume that $\overline b$$\overline b$ is an embedding and that

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$\overline b({\rm Int}D^2)$ misses $U\cup g(S^1\times S^{n-1})$$U\cup g(S^1\times S^{n-1})$.

Since $n-1>1$$n-1>1$, we have $\pi_1(V_{2n-2,n-1})=0$$\pi_1(V_{2n-2,n-1})=0$. Hence the standard $(n-1)$$(n-1)$-framing of $S^1\times1_{n-1}$$S^1\times1_{n-1}$ in $S^1\times S^{n-1}$$S^1\times S^{n-1}$ extends to an $(n-1)$$(n-1)$-framing on $\overline b(D^2)$$\overline b(D^2)$ in $\Rr^{2n}$$\Rr^{2n}$. Thus $\overline b$$\overline b$ extends to an embedding

$\displaystyle \widehat b:D^2\times D^{n-1}\to\R^{2n}-U\quad\text{such that}\quad \widehat b(\partial D^2\times D^{n-1})\subset g(S^1\times S^{n-1}).$ $$\displaystyle \widehat b:D^2\times D^{n-1}\to\R^{2n}-U\quad\text{such that}\quad \widehat b(\partial D^2\times D^{n-1})\subset g(S^1\times S^{n-1}).$$

Take an embedding $g_1:S^n\to\R^{2n}-U$$g_1:S^n\to\R^{2n}-U$ such that

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$$\displaystyle g_1(S^n)\ =\ g(S^1\times S^{n-1})-\widehat b(\partial D^2\times{\rm Int}D^{n-1}) \bigcup\limits_{\widehat b(\partial D^2\times\partial D^{n-1})} \widehat b(D^2\times\partial D^{n-1})\ \cong\ S^n.$$

with proper orientation so that $g_1$$g_1$ is homologous to $g$$g$. QED

The isotopy class of the embedding $f_a$$f_a$ is independent of the choises in the construction. The independence of the arc and of the maps $g,g_1$$g,g_1$ follows by $n\ge3$$n\ge3$ and by Proposition 4.3 below, respectively.

By Definition 5.1 of the Whitney invariant, $W(f_a,f_0)$$W(f_a,f_0)$ is $a$$a$ for $n\ge3$$n\ge3$ odd and $a \mod2$$a \mod2$ for $n\ge4$$n\ge4$ even. Thus by Theorem 2.1.a for $n\ge4$$n\ge4$ all isotopy classes of embeddings $N\to\Rr^{2n}$$N\to\Rr^{2n}$ can be obtained from any chosen embedding $f_0$$f_0$ by the above construction.

Proposition 4.3. For any $n\ge4$$n\ge4$ both the linked embedded connected sum and parametric connected sum (introduced in [Skopenkov2006a], [Skopenkov2015a]) define free transitive actions of $H_1(N;\Zz_{\varepsilon(n-1)})$$H_1(N;\Zz_{\varepsilon(n-1)})$ on $E^{2n}(N)$$E^{2n}(N)$.

This follows by Theorem 2.1.a and by [Skopenkov2014, Remark 18.a].

5 The Whitney invariant

Let $N$$N$ be a closed $n$$n$-manifold. Take an embedding $f_0:N\to\Rr^m$$f_0:N\to\Rr^m$. Fix an orientation on $\Rr^m$$\Rr^m$. For any other embedding $f \colon N \to \Rr^m$$f \colon N \to \Rr^m$ we define the Whitney invariant

$\displaystyle W(f, f_0)=W_{f_0}(f)=W(f)\in H_{2n-m+1}(N;\Zz_N).$ $$\displaystyle W(f, f_0)=W_{f_0}(f)=W(f)\in H_{2n-m+1}(N;\Zz_N).$$

Here the coefficients $\Zz_N$$\Zz_N$ are $\Zz$$\Zz$ if $N$$N$ is oriented and $m-n$$m-n$ is odd, and are $\Zz_2$$\Zz_2$ otherwise.

Roughly speaking,

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$W(f):=[{\rm Cl}\Sigma(H)]$ is defined as the homology class of the closure of the self-intersection set of a general position homotopy $H$$H$ between $f$$f$ and $f_0$$f_0$.

This is formalized in Definition 5.2 in the smooth category, following [Skopenkov2010], see also [Haefliger&Hirsch1963]. The definition in the PL category is analogous [Hudson1969, $\S$$\S$11], [Vrabec1977, p. 145], [Skopenkov2006, $\S$$\S$2.4 `The Whitney invariant']. We begin by presenting a simpler definition, Definition 5.1, for a particular case.

For Theorem 2.1 only the case $m=2n$$m=2n$ is required.

Definition 5.1. Assume that $N$$N$ is $(2n-m)$$(2n-m)$-connected and $2m\ge3n+3$$2m\ge3n+3$. Then by [Haefliger&Hirsch1963, Theorem 3.1.b] restrictions of $f$$f$ and $f_0$$f_0$ to $N_0$$N_0$ are isotopic, cf. [Takase2006, Lemma 2.2]. (Here is sketch of an argument. Using the Smale-Hirsch classification of immersions we obtain that restrictions of $f$$f$ and $f_0$$f_0$ to $N_0$$N_0$ are `regular homotopic', see [Koschorke2013, Definition 2.7]. Since $N$$N$ is $(2n-m)$$(2n-m)$-connected, $N_0$$N_0$ retracts to an $(m-n-1)$$(m-n-1)$-dimensional polyhedron. Therefore these restrictions are isotopic.)

So we can make an isotopy of $f$$f$ and assume that $f=f_0$$f=f_0$ on $N_0$$N_0$. Take a general position homotopy $F:B^n\times I\to\Rr^m$$F:B^n\times I\to\Rr^m$ relative to $\partial B^n$$\partial B^n$ between the restrictions of $f$$f$ and $f_0$$f_0$ to $B^n$$B^n$. Let $f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$$f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$ (`the intersection of this homotopy with $f(N-B^n)$$f(N-B^n)$').

Since $n+2(n+1)<2m$$n+2(n+1)<2m$, by general position

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${\rm Cl}(f\cap F)$ is a compact $(2n+1-m)$$(2n+1-m)$-manifold whose boundary is contained in $\partial N_0$$\partial N_0$.

So $f\cap F$$f\cap F$ carries a homology class with $\Zz_2$$\Zz_2$ coefficients. If $m-n$$m-n$ is odd and $N$$N$ is oriented, then $f\cap F$$f\cap F$ has a natural orientation defined below, and so carries a homology class with $\Zz$$\Zz$ coefficients. Define $W(f)$$W(f)$ to be the homology class:

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$$\displaystyle W(f):=[{\rm Cl}(f\cap F)]\in H_{2n-m+1}(N_0,\partial N_0;\Zz_N)\cong H_{2n-m+1}(N;\Zz_N).$$

The orientation on $f\cap F$$f\cap F$ (extendable to

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${\rm Cl}(f\cap F)$) is defined (for $m-n$$m-n$ odd and $N$$N$ is oriented) as follows (cf. Remark 7.3). For any point $x_f\in f\cap F$$x_f\in f\cap F$ take a base $\xi_f$$\xi_f$ at $x_f$$x_f$ tangent to $f\cap F$$f\cap F$. Complete this base to a positive base $(\xi_f,\eta_f)$$(\xi_f,\eta_f)$ tangent to $N$$N$. Since $n+2(n+1)<2m$$n+2(n+1)<2m$, by general position there is a unique point $x_F\in B^n\times I$$x_F\in B^n\times I$ such that $Fx_F=fx_f$$Fx_F=fx_f$. The tangent base $\xi_f$$\xi_f$ at $x_f$$x_f$ thus gives a base $\xi_F$$\xi_F$ at $x_F$$x_F$ tangent to $B^n\times I$$B^n\times I$ such that $df(x_f)\xi_f=dF(x_F)\xi_F$$df(x_f)\xi_f=dF(x_F)\xi_F$. Complete this base $\xi_F$$\xi_F$ to a positive base $(\xi_F,\eta_F)$$(\xi_F,\eta_F)$ tangent to $B^n\times I$$B^n\times I$, where the orientation on $B^n$$B^n$ comes from $N$$N$. The union $\zeta=(df(x_f)\xi_f,df(x_f)\eta_f,dF(x_F)\eta_F)$$\zeta=(df(x_f)\xi_f,df(x_f)\eta_f,dF(x_F)\eta_F)$ of the images of the constructed bases is a base at $fx_f=Fx_F$$fx_f=Fx_F$ of $\Rr^m$$\Rr^m$. If $\zeta$$\zeta$ is positive, then call the tangent base $\xi_f$$\xi_f$ of $f\cap F$$f\cap F$ `positive'. Since a change of the orientation on $f\cap F$$f\cap F$ forces a change of the orientation of $\zeta$$\zeta$, this condition indeed defines an orientation on $f\cap F$$f\cap F$.

Definition 5.2. Assume that $m\ge n+2$$m\ge n+2$. Take a general position homotopy $H:N\times I\to\Rr^m\times I$$H:N\times I\to\Rr^m\times I$ between $f_0$$f_0$ and $f$$f$.

The closure

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${\rm Cl}\Sigma(H)$ of the self-intersection set carries a cycle mod 2. If $N$$N$ is oriented and $m-n$$m-n$ is odd, the closure also carries an integer cycle. See [Hudson1967, $\S$$\S$11], [Skopenkov2006, $\S$$\S$2.3 `The Whitney obstruction']. Let us informally explain these facts. For $2m\ge3n+2$$2m\ge3n+2$ by general position the closure

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${\rm Cl}\Sigma(H)$ can be assumed to be a submanifold. In general, since $m\ge n+2$$m\ge n+2$, by general position the closure has codimension 2 singularities, as defined in $\S$$\S$7. So the closure carries a cycle mod 2. When $m-n$$m-n$ is odd the closure also has a canonical orientation (see Definition 7.1 and Remark 7.2), so the closure carries an integer cycle.

Define the Whitney invariant to be the homology class:

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$$\displaystyle W(f):=[{\rm Cl}\Sigma(H)]\in H_{2n-m+1}(N\times I;\Zz_N)\cong H_{2n-m+1}(N;\Zz_N).$$

Clearly, $W(f) = W(f')$$W(f) = W(f')$ if $f$$f$ is isotopic to $f'$$f'$. Hence the Whitney invariant defines a map

$\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_N),\quad [f] \mapsto W(f).$ $$\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_N),\quad [f] \mapsto W(f).$$

Clearly, $W(f_0)=0$$W(f_0)=0$ (for both definitions).

The definition of $W$$W$ depends on the choice of $f_0$$f_0$, but we write $W$$W$ not $W_{f_0}$$W_{f_0}$ for brevity.

Remark 5.3. (a) The Whitney invariant is well-defined by Definition 5.2, i.e. is independent of the choice of a general position homotopy $H:N\times I\to\Rr^m\times I$$H:N\times I\to\Rr^m\times I$ from $f_0$$f_0$ to $f$$f$.

This follows from the equality

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$[{\rm Cl}\Sigma(H_0)]−[{\rm Cl}\Sigma(H_1)] = \partial [{\rm Cl}\Sigma(H_{01})]$ for a general position homotopy $H_{01}:N\times I\times I\to\Rr^m\times I\times I$$H_{01}:N\times I\times I\to\Rr^m\times I\times I$ between general position homotopies $H_0,H_1:N\times I\to\Rr^m\times I$$H_0,H_1:N\times I\to\Rr^m\times I$ from $f_0$$f_0$ to $f$$f$. See details in [Hudson1969, $\S$$\S$11].

(b) Definition 5.1 is a particular case of Definition 5.2. (Indeed, if $f=f_0$$f=f_0$ on $N_0$$N_0$, we can take $H$$H$ to be fixed on $N_0$$N_0$. See details in [Skopenkov2010, Difference Lemma 2.4].) Hence the Whitney invariant is well-defined by Definition 5.1, i.e. independent of the choice of $F$$F$ and of the isotopy making $f=f_0$$f=f_0$ outside $B^n$$B^n$.

(c) The class $W(f)$$W(f)$ is independent of the choice of the orientation on $N$$N$ (because a change of the orientation on $N$$N$ forces a change of the orientation on $f\cap F$$f\cap F$ or on

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${\rm Cl}\Sigma(H)$). For the reflection $\sigma:\Rr^m\to\Rr^m$$\sigma:\Rr^m\to\Rr^m$ with respect to a hyperplane we have $W(\sigma\circ f)=-W(f)$$W(\sigma\circ f)=-W(f)$ (because a change of the orientation on $\Rr^m$$\Rr^m$ forces a change of the orientation on $f\cap F$$f\cap F$ or on

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${\rm Cl}\Sigma(H)$; for Definition 5.1 also observe that we may assume that $f=f_0=\sigma\circ f$$f=f_0=\sigma\circ f$ on $N_0$$N_0$).

(d) For the Hudson tori $W(\Hud_n(a))=W(\Hud'_n(a))$$W(\Hud_n(a))=W(\Hud'_n(a))$ is $a$$a$ or $a\mod2$$a\mod2$ for $n\ge3$$n\ge3$, and $W(\Hud'_2(a))=(a\mod2,0)$$W(\Hud'_2(a))=(a\mod2,0)$.

For $\Hud_n(a)$$\Hud_n(a)$ this is clear by Definition 5.1. For $\Hud'_n(a)$$\Hud'_n(a)$ and $n\ge3$$n\ge3$ this was proved in [Hudson1963] (with a different but equivalent definition of the Whitney invariant; using and proving a particular case of Remark 5.3.f). For $\Hud'_2(a)$$\Hud'_2(a)$ the proof is analogous.

(e) $W(f\#g)=W(f)$$W(f\#g)=W(f)$ for any pair of embeddings $f:N\to\Rr^m$$f:N\to\Rr^m$ and $g:S^n\to\Rr^m$$g:S^n\to\Rr^m$. This is clear by Definition 5.1 because $W(f\#g)-W(f)=W(f\#g,f_0)-W(f,f_0)=W(f\#g,f)=0$$W(f\#g)-W(f)=W(f\#g,f_0)-W(f,f_0)=W(f\#g,f)=0$. Let us prove the latter equality. Take the identical isotopy $H_f$$H_f$ of $f$$f$ and a general position homotopy $H_g$$H_g$ between $g$$g$ and the standard embedding. Then the boundary connected sum $H_f\sharp H_g$$H_f\sharp H_g$ is a general position homotopy between $f\#g$$f\#g$ and an embedding isotopic to $f$$f$. The cycle

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${\rm Cl}\Sigma(H_f\sharp H_g)={\rm Cl}\Sigma H_g$ is null-homologous in $S^n$$S^n$ and hence in $N\cong N\#S^n$$N\cong N\#S^n$; cf. [Skopenkov2008, Addendum to the Classification Theorem].

(f) For $m=2n+1$$m=2n+1$ and $N=S^n\sqcup S^n$$N=S^n\sqcup S^n$ the Whitney invariant equals to the pair of linking coefficients [Skopenkov2016h, $\S$$\S$3].

(g) The Whitney invariant need not be a bijection for $m<2n$$m<2n$. This is seen, for example, by applying Theorem 6.4 below in case of knotted tori [Skopenkov2016k, Theorem 5.1]) or by taking $n$$n$ even, $N$$N$ non-orientable, $m=2n$$m=2n$ and applying by Theorem 2.1.b.

(h) The paper [Haefliger&Hirsch1963, $\S$$\S$2] gives yet another equivalent definition of the Whitney invariant, and proves its injectivity under certain restrictions (the Haefliger-Hirsch $h$$h$-principle, see Theorem 6.2).

(i) Let $N$$N$ be a closed orientable $n$$n$-manifold. For two embeddings $f,g:N\to\R^{2n-1}$$f,g:N\to\R^{2n-1}$ the invariant defined in [Haefliger&Hirsch1963, $\S$$\S$2] equals plus or minus their Whitney invariant $W(f,g)$$W(f,g)$ from Definition 5.1. This is proved analogously to the following version.

Let $D^n$$D^n$ and $B^n$$B^n$ be closed $n$$n$-balls in $N$$N$ such that

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$D^n\subset{\rm Int}B^n$. Let $f:N\to\R^{2n-1}$$f:N\to\R^{2n-1}$ a general position immersion which is an embedding on the closure of $N-D^n$$N-D^n$. Let $N_0 := N-B^n$$N_0 := N-B^n$ and $f_0:=f|_{N-B^n}$$f_0:=f|_{N-B^n}$. We omit $\Z_{(n)}$$\Z_{(n)}$-coefficients. Then $f_0^{-1}f(D^n)$$f_0^{-1}f(D^n)$ represents a 1-cycle in $N$$N$. The homology class of this 1-cycle equals minus the normal Stiefel-Whitney class $PD\overline{W}^{n-1}(N)\in H_1(N)$$PD\overline{W}^{n-1}(N)\in H_1(N)$. This follows because

$\displaystyle 0 \overset{(1)}= [f^{-1}f'(N)] \overset{(2)}= [f_0^{-1}f'(N_0)]+[f_0^{-1}f'(B^n)] \overset{(3)} = PD\overline{W}^{n-1}(N)+[f_0^{-1}f(D^n)].$ $$\displaystyle 0 \overset{(1)}= [f^{-1}f'(N)] \overset{(2)}= [f_0^{-1}f'(N_0)]+[f_0^{-1}f'(B^n)] \overset{(3)} = PD\overline{W}^{n-1}(N)+[f_0^{-1}f(D^n)].$$

Here

• $f':N\to\R^{2n-1}$$f':N\to\R^{2n-1}$ is the map formed by a general position small length vector field normal to $f$$f$, and non-zero over $f(B^n)$$f(B^n)$;

• equality (1) holds because $f'$$f'$ is homotopic to a map whose image is disjoint from $f$$f$;

• equality (2) holds because $f_0$$f_0$-preimages of both $f'(N_0)$$f'(N_0)$ and $f'B^n$$f'B^n$ are 1-cycles, and $f|_{B^n}$$f|_{B^n}$-preimage of $f'(N)$$f'(N)$ is a 1-cycle which is null-homologous;

• equality (3) holds because the Euler class of a bundle is dual to the homology class of zeroes of a general position section, and because $[f_0^{-1}f'(B^n)] = [f_0^{-1}f'(D^n)] \overset{(4)}= [f_0^{-1}f(D^n)]$$[f_0^{-1}f'(B^n)] = [f_0^{-1}f'(D^n)] \overset{(4)}= [f_0^{-1}f(D^n)]$;

• equality (4) holds because $f(D^n)$$f(D^n)$ is homologous relative to the boundary and outside $f(N_0)$$f(N_0)$ to the union of $f'(D^n)$$f'(D^n)$ and the the normal vectors at points of $f(\partial D^n)$$f(\partial D^n)$.

6 A generalization to highly-connected manifolds

In this section let $N$$N$ be a closed orientable homologically $k$$k$-connected $n$$n$-manifold, $k\ge0$$k\ge0$. Recall the unknotting theorem [Skopenkov2016c, Theorem 2.4] that all embeddings $N \to\Rr^m$$N \to\Rr^m$ are isotopic when $m\ge 2n-k+1$$m\ge 2n-k+1$ and $n\ge2k+2$$n\ge2k+2$. In this section we generalize Theorem 2.1 to a description of $E^{2n-k}(N)$$E^{2n-k}(N)$ and further to $E^m(N)$$E^m(N)$ for $m\ge2n-2k+1$$m\ge2n-2k+1$.

6.1 Examples

Some simple examples are the Hudson tori $\Hud_{m,n,p}(a), \Hud'_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$$\Hud_{m,n,p}(a), \Hud'_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$.

Example 6.1 (cf. [Skopenkov2010, Definition 1.4]). Assume that $N$$N$ is $k$$k$-connected and $n-k\ge3$$n-k\ge3$. Then for an embedding $f_0:N\to\Rr^{2n-k}$$f_0:N\to\Rr^{2n-k}$ and a class $a\in H_{k+1}(N;\Zz)$$a\in H_{k+1}(N;\Zz)$ one can construct an embedding $f_a:N\to\Rr^{2n-k}$$f_a:N\to\Rr^{2n-k}$ by linked embedded connected sum analogously to the case $k=0$$k=0$ presented in Example 4.1.

We have $W(f_a,f_0)=a$$W(f_a,f_0)=a$ for the Whitney invariant. Hence by Theorem 6.2 below this construction gives a free transitive action of $H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$$H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$ on $E^{2n-k}(N)$$E^{2n-k}(N)$, provided $n\ge k+3$$n\ge k+3$ or $n\ge2k+4$$n\ge2k+4$ in the PL or smooth categories, respectively.

The embedding $f_a$$f_a$ has an alternative construction using parametric connected sum [Skopenkov2014, Remark 18.a].

6.2 Classification

Theorem 6.2. Let $N$$N$ be a closed oriented homologically $k$$k$-connected $n$$n$-manifold, $k\ge0$$k\ge0$. Then the Whitney invariant

$\displaystyle W:E^{2n-k}(N)\to H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$ $$\displaystyle W:E^{2n-k}(N)\to H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$$

is a bijection, provided $n\ge2k+4$$n\ge2k+4$ in the smooth category or $n\ge k+3$$n\ge k+3$ in the PL category.

This was proved for $k$$k$-connected manifolds in the smooth category [Haefliger&Hirsch1963, Theorem 2.4], and in the PL category in [Weber1967], [Hudson1969, $\S$$\S$11], cf. [Boechat&Haefliger1970, Theorem 1.6], [Boechat1971, Theorem 4.2], [Vrabec1977, Theorems 1.1 and 1.2], [Adachi1993, $\S$$\S$7]. The proof actually used the homological $k$$k$-connectedness assumption (basically because the $k$$k$-connectedness was used to ensure high enough connectedness of the complement in $\Rr^m$$\Rr^m$ to the image of $N$$N$, by Alexander duality and simple connectedness of the complement, so homological $k$$k$-connectedness of $N$$N$ is sufficient).

For $k=0$$k=0$ Theorem 6.2 is covered by Theorem 2.1; for $k\ge1$$k\ge1$ it is not. For $k+3\le n\le2k+1$$k+3\le n\le2k+1$ the PL case of Theorem 6.2 gives nothing but the Zeeman Unknotting Spheres Theorem [Skopenkov2016c, Theorem 2.3]. For the case of knotted tori see [Skopenkov2016k, Theorem 3.1].

An inverse to the map $W$$W$ of Theorem 6.2 is given by Example 6.1.

Because of the existence of knotted spheres the analogues of Theorem 6.2 for $n=k+2$$n=k+2$ in the PL case, and for $n\le2k+3$$n\le2k+3$ in the smooth case are false. So for the smooth category, $n\le2k+3$$n\le2k+3$ and $N$$N$ closed connected, a classification of $E^{2n-k}(N)$$E^{2n-k}(N)$ is much harder: for 40 years the only known complete readily calculable classification results were for homology spheres $N$$N$. E.g. $E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)}$$E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)}$ for any $s\ge2$$s\ge2$ [Haefliger1966, Corollary 8.14], [Skopenkov2016s, Theorem 3.2]. The following result for $n=2k+3$$n=2k+3$ was obtained using the Boéchat-Haefliger formula for the smoothing obstruction [Boechat&Haefliger1970, Theorem 2.1], [Boechat1971, Theorem 5.1]. Using that formula one can define the higher-dimensional Kreck invariant [Skopenkov2008, $\S$$\S$4].

Theorem 6.3 [Skopenkov2008, Higher-dimensional Classification Theorem]. Let $N$$N$ be a closed orientable homologically $(2l-2)$$(2l-2)$-connected $(4l-1)$$(4l-1)$-manifold. Then the Whitney invariant

$\displaystyle W:E^{6l}_D(N)\to H_{2l-1}(N)$ $$\displaystyle W:E^{6l}_D(N)\to H_{2l-1}(N)$$

is surjective and for any $u\in H_{2l-1}(N)$$u\in H_{2l-1}(N)$ the Kreck invariant

$\displaystyle \eta_u:W^{-1}u\to\Zz_{d(u)}$ $$\displaystyle \eta_u:W^{-1}u\to\Zz_{d(u)}$$

is a 1-1 correspondence, where $d(u)$$d(u)$ is the divisibility of the projection of $u$$u$ to the free part of $H_1(N)$$H_1(N)$.

Recall that the divisibility of zero is zero and the divisibility of $x\in G-\{0\}$$x\in G-\{0\}$ is $\max\{d\in\Zz\ | \ \text{there is }x_1\in G: \ x=dx_1\}$$\max\{d\in\Zz\ | \ \text{there is }x_1\in G: \ x=dx_1\}$.

How does one describe $E^m(N)$$E^m(N)$ when $N$$N$ is not $(2n-m)$$(2n-m)$-connected? For general $N$$N$ see the sentence on $E^{2n-1}(N)$$E^{2n-1}(N)$ at the end of $\S$$\S$2. We can say more as the connectivity $k$$k$ of $N$$N$ increases. Some estimations of $E^{2n-k-1}(N)$$E^{2n-k-1}(N)$ for a closed $k$$k$-connected $n$$n$-manifold $N$$N$ are presented in [Skopenkov2010]. For $k>1$$k>1$ one can go even further:

Theorem 6.4 [Becker&Glover1971, Corollary 1.3]. Let $N$$N$ be a closed $k$$k$-connected $n$$n$-manifold embeddable into $\Rr^m$$\Rr^m$, $m\ge2n-2k+1$$m\ge2n-2k+1$ and $2m\ge 3n+4$$2m\ge 3n+4$. Then there is a 1-1 correspondence

$\displaystyle E^m(N)\to [N_0, V_{m,n+1}].$ $$\displaystyle E^m(N)\to [N_0, V_{m,n+1}].$$

The 1-1 correspondence is constructed using `the Haefliger-Wu invariant' involving the configuration space of distinct pairs [Skopenkov2006, $\S$$\S$5]. For $k=0$$k=0$ Theorem 6.4 is the same as General Position Theorem [Skopenkov2016c, Theorem 2.1] (because $V_{2n+1,n+1}$$V_{2n+1,n+1}$ is $(n-1)$$(n-1)$-connected). For $k=1$$k=1$ Theorem 6.4 is covered by Theorem 6.2; for $k\ge2$$k\ge2$ it is not. For application to knotted tori see [Skopenkov2016k, Theorem 5.1]. For generalization to arbitrary manifolds see survey [Skopenkov2006, $\S$$\S$5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Observe that in Theorem 6.4 $V_{m,n+1}$$V_{m,n+1}$ can be replaced by $V_{M,M+n-m+1}$$V_{M,M+n-m+1}$ for any $M>n$$M>n$.

7 An orientation on the self-intersection set

Let $f:N\to\Rr^m$$f:N\to\Rr^m$ be a smooth map from an oriented $n$$n$-manifold $N$$N$ where $m\ge n+2$$m\ge n+2$. We assume that the closure

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${\rm Cl}\Sigma(f)$ of the self-intersection set of $f$$f$ has codimension 2 singularities, i.e., there is

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$P\subset{\rm Cl}\Sigma(f)$ such that

• both $P$$P$ and

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  ${\rm Cl}\Sigma(f)$ are subpolyhedra of some triangulation of $N$$N$,
• we have $\dim P\le\dim\Sigma(f)-2$$\dim P\le\dim\Sigma(f)-2$ and
• $\Sigma(f)-P$$\Sigma(f)-P$ is an open manifold consisting of self-transverse double points of $f$$f$.

Definition 7.1 (A canonical orientation on $\Sigma(f)-P$$\Sigma(f)-P$). Take points $x,y\in N$$x,y\in N$ away from $P$$P$ and such that $fx=fy$$fx=fy$. Then a $(2n-m)$$(2n-m)$-base $\xi_x$$\xi_x$ tangent to $\Sigma(f)-P$$\Sigma(f)-P$ at $x$$x$ gives a $(2n-m)$$(2n-m)$-base $\xi_y:=df_y^{-1}df_x(\xi_x)$$\xi_y:=df_y^{-1}df_x(\xi_x)$ tangent to $\Sigma(f)-P$$\Sigma(f)-P$ at $y$$y$. Since $N$$N$ is oriented, we can take positive $(m-n)$$(m-n)$-bases $\eta_x$$\eta_x$ and $\eta_y$$\eta_y$ at $x$$x$ and $y$$y$ normal to $\xi_x$$\xi_x$ and to $\xi_y$$\xi_y$. If the base $(df_x(\xi_x),df_x(\eta_x),df_y(\eta_y))$$(df_x(\xi_x),df_x(\eta_x),df_y(\eta_y))$ of $\Rr^m$$\Rr^m$ is positive, then call the base $\xi_x$$\xi_x$ positive. This is well-defined because a change of the sign of $\xi_x$$\xi_x$ forces changes of the signs of $\xi_y,\eta_x$$\xi_y,\eta_x$ and $\eta_y$$\eta_y$.

Remark 7.2 (Properties of the orientation just defined on $\Sigma(f) - P$$\Sigma(f) - P$)..

1. A change of the orientation of $N$$N$ forces changes of the signs of $\eta_x$$\eta_x$ and $\eta_y$$\eta_y$ and so does not change the orientation of $\Sigma(f)-P$$\Sigma(f)-P$.
2. The orientation on $\Sigma(f)-P$$\Sigma(f)-P$ need not extend to

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   ${\rm Cl}\Sigma(f)$: take the smooth cone $f:D^3\to\Rr^5$$f:D^3\to\Rr^5$ over a general position map $g:S^2\to\Rr^4$$g:S^2\to\Rr^4$ having only two transverse self-intersection points, where the smooth cone is defined by $f(tx):=(g(x)\sin(\pi t/2),\cos(\pi t/2))$$f(tx):=(g(x)\sin(\pi t/2),\cos(\pi t/2))$, for $x\in S^2$$x\in S^2$ and $t\in[0,1]$$t\in[0,1]$.
3. The orientation on $\Sigma(f)-P$$\Sigma(f)-P$ extends to

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   ${\rm Cl}\Sigma(f)$ if $m-n$$m-n$ is odd [Hudson1969, Lemma 11.4].

Remark 7.3 (A canonical orientation on $f\Sigma(f)-fP$$f\Sigma(f)-fP$ for $m-n$$m-n$ even). This remark is added as a complement for Definition 7.1 but is not needed for the definition of the Whitney invariant.

Take a $(2n-m)$$(2n-m)$-base $\xi$$\xi$ at a point $x\in f\Sigma(f)-fP$$x\in f\Sigma(f)-fP$. Since $N$$N$ is oriented, we can take a positive $(m-n)$$(m-n)$-base $\eta_+$$\eta_+$ normal to $f\Sigma(f)$$f\Sigma(f)$ in one sheet of $fN$$fN$. Analogously construct an $(m-n)$$(m-n)$-base $\eta_-$$\eta_-$ for the other sheet of $fN$$fN$. Since $m-n$$m-n$ is even, the orientation of the base $(\xi,\eta_+,\eta_-)$$(\xi,\eta_+,\eta_-)$ of $\Rr^m$$\Rr^m$ does not depend on choosing the first and the other sheet of $fN$$fN$ at $x$$x$. If the base $(\xi,\eta_+,\eta_-)$$(\xi,\eta_+,\eta_-)$ is positive, then call the base $\xi$$\xi$ positive. This is well-defined because a change of the sign of $\xi$$\xi$ forces changes of the signs of $\eta_+,\eta_-$$\eta_+,\eta_-$ and so of $(\xi,\eta_+,\eta_-)$$(\xi,\eta_+,\eta_-)$.

We remark that a change of the orientation of $N$$N$ forces changes of the signs of $\eta_+,\eta_-$$\eta_+,\eta_-$ and so does not change the orientation of $f\Sigma(f)-fP$$f\Sigma(f)-fP$.

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