Embeddings just below the stable range: classification

This page has been accepted for publication in the Bulletin of the Manifold Atlas.

1 Introduction

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

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].

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, either $n$$n$ is odd or $N$$N$ is orientable, and either $n\ge4$$n\ge4$ or $n=3$$n=3$ and we are in the PL category. 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.

This is proved in [Haefliger1962b, 1.3.e], [Haefliger1963], [Haefliger&Hirsch1963, Theorem 2.4], [Bausum1975, Theorem 43] in the smooth category, and in [Weber1967], [Vrabec1977, Theorem 1.1] in the PL category.

If $n\ge4$$n\ge4$ is even and $N$$N$ is a closed connected non-orientable $n$$n$-manifold, then [Bausum1975, Theorem 43] asserts that there is a 1-1 correspondence

$\displaystyle E^{2n}(N)\to \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s$ $$\displaystyle E^{2n}(N)\to \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s$$

(because $Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$$Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$ is given by multiplication with $w_1(N)$$w_1(N)$). This 1-1 correspondence can presumably be defined as a generalized Whitney invariant, but the proof used the Haefliger-Wu invariant whose definition can be found in [Skopenkov2006, $\S$$\S$5]. It would be interesting to check if this description of $E^{2n}(N)$$E^{2n}(N)$ is equivalent to its different forms [Haefliger1962b, 1.3.e], [Haefliger1963], [Vrabec1977, Theorem 1.1].

The classification of smooth embeddings of 3-manifolds in $\Rr^6$$\Rr^6$ is more complicated, see $\S$$\S$6.3 or [Skopenkov2016t].

For embeddings of $n$$n$-manifolds in $\Rr^{2n-1}$$\Rr^{2n-1}$ see the case of 4-manifolds [Skopenkov2016f], [Yasui1984] for $n\ge5$$n\ge5$ and [Saeki1999], [Skopenkov2010], [Tonkonog2010] for non-closed manifolds.

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

3 Hudson tori

Together with the Haefliger knotted sphere [Skopenkov2016t], the examples of Hudson tori presented below were the first examples of embeddings in codimension greater than 2 which are not isotopic to the standard embedding defined below. (Hudson's construction [Hudson1963] was not as explicit as those below.)

For $m\ge n+2$$m\ge n+2$ we define the standard embedding $S^p\times S^{n-p}\to\Rr^m$$S^p\times S^{n-p}\to\Rr^m$ as the composition of the standard embeddings $S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}=\Rr^{n+2}\to\Rr^m$$S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}=\Rr^{n+2}\to\Rr^m$. (Note that in [Skopenkov2015], [Skopenkov2015a] a different embedding is called a standard embedding. Since these two embeddings are isotopic, no confusion would appear.)

Let $1_n:=(1,0,\ldots,0)\in S^n$$1_n:=(1,0,\ldots,0)\in S^n$.

Let us construct, for each $a\in\Zz$$a\in\Zz$ and $n\ge2$$n\ge2$, an embedding

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

The reader might first consider the case $n=2$$n=2$.

Definition 3.1. Take the standard embeddings $2D^{n+1}\times S^{n-1}\subset\Rr^{2n}$$2D^{n+1}\times S^{n-1}\subset\Rr^{2n}$ (where $2$$2$ means homothety with coefficient 2) and $\partial D^2\subset\partial D^{n+1}$$\partial D^2\subset\partial D^{n+1}$. Take the embedded sphere and embedded torus,

$\displaystyle 2\partial D^{n+1}\times-1_{n-1}\subset 2D^{n+1}\times S^{n-1}\subset\Rr^{2n}\quad\text{and}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$ $$\displaystyle 2\partial D^{n+1}\times-1_{n-1}\subset 2D^{n+1}\times S^{n-1}\subset\Rr^{2n}\quad\text{and}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$

Join them by an arc whose interior misses the two embedded manifolds. The Hudson torus $\Hud_n(1)$$\Hud_n(1)$ is the embedded connected sum of the two embeddings along this arc, compatible with the orientation. (Unlike the unlinked embedded connected sum [Skopenkov2016c] this is a linked embedded connected sum, i.e. the connected sum of two embeddings whose images are not contained in disjoint cubes.)

We remark that the construction in Definition 3.1 works for $n=1$$n=1$. This is not the case for the next construction in Definition 3.2.

Definition 3.2. For $a\in\Zz$$a\in\Zz$ we repeat the construction of Definition 3.1 replacing $2\partial D^{n+1}\times-1_{n-1}$$2\partial D^{n+1}\times-1_{n-1}$ by $|a|$$|a|$ copies $(1+\frac1k)\partial D^{n+1}\times-1_{n-1}$$(1+\frac1k)\partial D^{n+1}\times-1_{n-1}$ ($k=1,\dots,|a|$$k=1,\dots,|a|$) of $S^n$$S^n$. The copies are outside $D^{n+1}\times S^{n-1}$$D^{n+1}\times S^{n-1}$ and are `parallel' to $\partial D^{n+1}\times-1_{n-1}$$\partial D^{n+1}\times-1_{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 by tubes joining each $k$$k$-th copy to the $(k+1)$$(k+1)$-th copy. We obtain an embedding $g:S^n\to\Rr^{2n}$$g:S^n\to\Rr^{2n}$. Let $\Hud_n(a)$$\Hud_n(a)$ be the linked embedded connected sum of $g$$g$ with the embedding $\partial D^2\times S^{n-1}\subset\Rr^{2n}$$\partial D^2\times S^{n-1}\subset\Rr^{2n}$ from Definition 3.1.

Clearly, $\Hud_n(0)$$\Hud_n(0)$ is isotopic to the standard embedding.

The original motivation for Hudson was that $\Hud_n(1)$$\Hud_n(1)$ is not isotopic to $\Hud_n(0)$$\Hud_n(0)$ for each $n$$n$ (this is a particular case of Proposition 3.3 below).

One guesses 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 exists and is `realized' by the homotopy class of the map

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

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

Proposition 3.3. 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 3.3 follows by calculation of the Whitney invariant (Remark 5.3.e below) and, for $n$$n$ even, by Theorem 2.1.

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)$ isotopic to $\Hud_2(2)$$\Hud_2(2)$? It would also be interesting to find an explicit construction of an isotopy between $\Hud_{2k}(a)$$\Hud_{2k}(a)$ and $\Hud_{2k}(a+2)$$\Hud_{2k}(a+2)$, cf. [Vrabec1977, $\S$$\S$5].

Definition 3.4. Let us give, for $a\in\Zz$$a\in\Zz$ and $n\ge2$$n\ge2$, another construction of embeddings

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

Define a map $S^0\times S^{n-1}\to D^n$$S^0\times S^{n-1}\to D^n$ to be the constant $0\in D^n$$0\in D^n$ on one component $1\times S^{n-1}$$1\times S^{n-1}$ and the `standard inclusion' $\{-1\}\times S^{n-1}\to\partial D^n\subset D^n$$\{-1\}\times S^{n-1}\to\partial D^n\subset D^n$ on the other component. This map gives an embedding

$\displaystyle S^0\times S^{n-1}\to D^n\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$ $$\displaystyle S^0\times S^{n-1}\to D^n\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$

(See [Skopenkov2006, Figure 2.2]. The image of this embedding is the union of the standard $S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$$S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$ and the graph of the identity map in $S^{n-1}\times S^{n-1}\subset D^n\times S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$$S^{n-1}\times S^{n-1}\subset D^n\times S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$.)

Take any $x\in S^{n-1}$$x\in S^{n-1}$. The disk $D^{n+1}\times x$$D^{n+1}\times x$ intersects the image of this embedding at two points lying in $D^n\times x$$D^n\times x$, i.e., at the image of an embedding $S^0\times x\to D^n\times x$$S^0\times x\to D^n\times x$. Extend the latter embedding to an embedding $S^1\times x\to D^{n+1}\times x$$S^1\times x\to D^{n+1}\times x$. See [Skopenkov2006, Figure 2.3].) Thus we obtain the Hudson torus

$\displaystyle \Hud_n'(1):S^1\times S^{n-1}\overset h\to D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$ $$\displaystyle \Hud_n'(1):S^1\times S^{n-1}\overset h\to D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$

Here $h(e^{i\theta},y):=(y\cos\theta,\sin\theta,y)$$h(e^{i\theta},y):=(y\cos\theta,\sin\theta,y)$, where $D^{n+1}$$D^{n+1}$ is identified with $D^n\times D^1$$D^n\times D^1$.

The embedding $\Hud_n'(a)$$\Hud_n'(a)$ is obtained in the same way starting from a map $\varphi:\{-1\}\times S^{n-1}\to\partial D^n$$\varphi:\{-1\}\times S^{n-1}\to\partial D^n$ of degree $a$$a$ instead of the `standard inclusion'.

Remark 3.5. (a) The analogue of Proposition 3.3 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$$n\ge4$ and are PL isotopic for $n\ge3$$n\ge3$ [Skopenkov2006a]. It would be interesting to know if they are isotopic for $n=2$$n=2$, or if they are smoothly isotopic for $n=3$$n=3$.

(c) For $n=2$$n=2$ these construction give what we call the left Hudson torus. The right Hudson torus is constructed analogously. It is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of $S^1\times S^1$$S^1\times S^1$. The right and the left Hudson tori are not isotopic by Remark 5.3.e 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$ 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})$. There are versions of these constructions corresponding to Definition 3.4. For $p=0$$p=0$ this corresponds to the Zeeman construction [Skopenkov2016h] and its composition with the second unframed Kirby move. It would be interesting to know if the links $\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$$\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$ are isotopic, cf. [Skopenkov2015a, Remark 2.9.b]. These constructions can be further generalized [Skopenkov2016k].

4 Action by linked embedded connected sum

In this section we generalize the construction of the Hudson torus $\Hud(a)$$\Hud(a)$. For $n\ge3$$n\ge3$, a closed connected orientable $n$$n$-manifold $N$$N$, an embedding $f_0:N\to\Rr^{2n}$$f_0:N\to\Rr^{2n}$ and $a\in H_1(N;\Zz_{\varepsilon(n-1)})$$a\in H_1(N;\Zz_{\varepsilon(n-1)})$, we construct an embedding $f_a:N\to\Rr^{2n}$$f_a:N\to\Rr^{2n}$. This embedding is obtained by linked embedded connected sum of $f_0$$f_0$ with an $n$$n$-sphere representing the homology Alexander dual of $a$$a$.

More precisely, represent $a$$a$ by an embedding $a:S^1\to N$$a:S^1\to N$. 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}$. Identify $\nu_{f_0}^{-1}a(S^1)$$\nu_{f_0}^{-1}a(S^1)$ with $S^1\times S^{n-1}$$S^1\times S^{n-1}$. In the next paragraph we recall definition of embedded surgery on $S^1\times*\subset S^1\times S^{n-1}$$S^1\times*\subset S^1\times S^{n-1}$, which yields an embedding $g:S^n\to C_{f_0}$$g:S^n\to C_{f_0}$. Then we define $f_a$$f_a$ to be the (linked) embedded connected sum of $f_0$$f_0$ and $g$$g$ (along a certain arc joining their images).

Take a vector field on $S^1\times-1_{n-1}$$S^1\times-1_{n-1}$ normal to $S^1\times S^{n-1}$$S^1\times S^{n-1}$. Extend $S^1\times-1_{n-1}$$S^1\times-1_{n-1}$ along this vector field to a map $\overline a:D^2\to\Rr^{2n}$$\overline a:D^2\to\Rr^{2n}$. Since $2n>4$$2n>4$ and $n+2<2n$$n+2<2n$, by general position we may assume that $\overline a$$\overline a$ is an embedding and $\overline a(Int D^2)$$\overline a(Int D^2)$ misses $f_0(N)\cup S^1\times S^{n-1}$$f_0(N)\cup S^1\times S^{n-1}$. Since $n-1>1$$n-1>1$, we have $\pi_1(V_{2n-2,n-1})=0$$\pi_1(V_{2n-2,n-1})=0$. Hence the standard framing of $S^1\times-1_{n-1}$$S^1\times-1_{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 a(D^2)$$\overline a(D^2)$ in $\Rr^{2n}$$\Rr^{2n}$. Thus $\overline a$$\overline a$ extends to an embedding

$\displaystyle \widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}.$ $$\displaystyle \widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}.$$

Define an embedding $g:S^n\to C_{f_0}$$g:S^n\to C_{f_0}$ by setting

$\displaystyle g(S^n):\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n$ $$\displaystyle g(S^n):\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n$$

with natural orientation.

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

Proposition 4.1. Linked embedded connected sum, or parametric connected sum, 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)$, unless $n=3$$n=3$ in the smooth category.

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

5 The Whitney invariant

Let $N$$N$ be a closed $n$$n$-manifold and fix an embedding $f_0:N\to\Rr^m$$f_0:N\to\Rr^m$. For any other embedding $f \colon N \to \Rr^m$$f \colon N \to \Rr^m$, there is an invariant $W(f, f_0)$$W(f, f_0)$, called the Whitney invariant, which we define in this section. Roughly speaking, $W(f,f_0)=W_{f_0}(f)=W(f)$$W(f,f_0)=W_{f_0}(f)=W(f)$ is defined as the homology class of the self-intersection set $\Sigma(H)$$\Sigma(H)$ of a general position homotopy $H$$H$ between $f$$f$ and $f_0$$f_0$. This is formalized in Definition 5.2 in the smooth category, following [Skopenkov2010]. The definition in the PL category is analogous [Hudson1969, $\S$$\S$12], [Vrabec1977, p. 145], [Skopenkov2006, $\S$$\S$2.4]. We begin by presenting a simpler definition of the Whitney invariant, Definition 5.1, for a particular case. For Theorem 2.1 only the case $m=2n$$m=2n$ is required.

Fix an orientation on $\Rr^m$$\Rr^m$. Assume that either $m-n$$m-n$ is even or $N$$N$ is oriented.

Definition 5.1. Assume that $N$$N$ is $(2n-m)$$(2n-m)$-connected and $2m\ge3n+3$$2m\ge3n+3$. Then restrictions of $f$$f$ and $f_0$$f_0$ to $N_0$$N_0$ are regular homotopic [Hirsch1959]. 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, cf. [Haefliger&Hirsch1963, 3.1.b], [Takase2006, Lemma 2.2]. 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$$n+2(n+1)<2m$, by general position $Cl(f\cap F)$$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. For $m-n$$m-n$ odd it has a natural orientation defined below, and so carries a homology class with $\Zz$$\Zz$ coefficients. Define $W(f)$$W(f)$ to be the homology class:

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

The orientation on $f\cap F$$f\cap F$ (extendable to $Cl(f\cap F)$$Cl(f\cap F)$) is defined for $m-n$$m-n$ odd as follows. For each point $x\in f\cap F$$x\in f\cap F$ take a vector at $x$$x$ tangent to $f\cap F$$f\cap F$. Complete this vector to a positive base tangent to $N$$N$. Since $n+2(n+1)<2m$$n+2(n+1)<2m$, by general position there is a unique point $y\in B^n\times I$$y\in B^n\times I$ such that $Fy=fx$$Fy=fx$. The tangent vector at $x$$x$ thus gives a tangent vector at $y$$y$ to $B^n\times I$$B^n\times I$. Complete this vector to a positive base tangent to $B^n\times I$$B^n\times I$, where the orientation on $B^n$$B^n$ comes from $N$$N$. The union of the images of the constructed two bases is a base at $Fy=fx$$Fy=fx$ of $\Rr^m$$\Rr^m$. If the latter base is positive, then call the initial vector 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 the latter base of $\Rr^m$$\Rr^m$, this condition indeed defines an orientation on $f\cap F$$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$. Since $m\ge n+2$$m\ge n+2$, by general position the closure $Cl\Sigma(H)$$Cl\Sigma(H)$ of the self-intersection set has codimension 2 singularities. So the closure carries a homology class with $\Zz_2$$\Zz_2$ coefficients. (Note that $Cl\Sigma(H)$$Cl\Sigma(H)$ can be assumed to be a submanifold for $2m\ge3n+2$$2m\ge3n+2$.) For $m-n$$m-n$ odd it has a natural orientation ($\S$$\S$7) and so carries a homology class with $\Zz$$\Zz$ coefficients. Define the Whitney invariant to be the homology class:

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

(The orientation is defined for each $m,n$$m,n$ but used only for odd $m-n$$m-n$. When $m-n$$m-n$ is even, for $W(f)$$W(f)$ being well-defined we need $\Zz_2$$\Zz_2$-coefficients.)

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

$\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_{\varepsilon(m-n-1)}),\quad [f] \mapsto W(f) = W(f, f_0).$ $$\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_{\varepsilon(m-n-1)}),\quad [f] \mapsto W(f) = W(f, f_0).$$

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 $H$$H$, which is proved analogously to [Skopenkov2006, $\S$$\S$2.4].

(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 $F$$F$ to be fixed on $N_0$$N_0$.) 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) Since a change of the orientation on $N$$N$ forces a change of the orientation on $B^n$$B^n$, the class $W(f)$$W(f)$ is independent of the choice of the orientation on $N$$N$. 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 we may assume that $f=f_0=\sigma\circ f$$f=f_0=\sigma\circ f$ on $N_0$$N_0$ and because a change of the orientation of $\Rr^m$$\Rr^m$ forces a change of the orientation of $f\cap F$$f\cap F$).

(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$, is $(a\mod2,0)$$(a\mod2,0)$ for $n=2$$n=2$.

(e) $W(f\#g)=W(f)$$W(f\#g)=W(f)$ for each pair of embeddings $f:N\to\Rr^m$$f:N\to\Rr^m$ and $g:S^n\to\Rr^m$$g:S^n\to\Rr^m$.

(f) For $m=2n+1$$m=2n+1$ the Whitney invariant can be recovered from the collection of pairwise linking coefficients of the components of $N$$N$, cf. Remark 3.2.b of [Skopenkov2016h].

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] 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 (Linked embedded connected sum, cf. [Skopenkov2010, Definition 1.4]). If $N$$N$ is $k$$k$-connected, then for an embedding $f_0:N\to S^{2n-k}$$f_0:N\to S^{2n-k}$ and a class $a\in H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$$a\in H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$ one can construct an embedding $f_a:N\to S^{2n-k}$$f_a:N\to S^{2n-k}$ by linked connected sum analogously to the case $k=0$$k=0$.

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$$n\ge2k+4$ in the PL or smooth categories, respectively). If $n=2k+3$$n=2k+3$, then this construction gives only a construction of embeddings $f_a:N\to\Rr^{2n-k}$$f_a:N\to\Rr^{2n-k}$ for each $a\in H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$$a\in H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$ but not a well-defined action of $H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$$H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$ on $E^{2n-k}(N)$$E^{2n-k}(N)$.

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

6.2 Classification just below the stable range

Theorem 6.2. Let $N$$N$ be a closed orientable homologically $k$$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\ge k+3$$n\ge k+3$ in the smooth category or $n\ge2k+4$$n\ge2k+4$ in the categoriy.

This was proved for $k$$k$-connected manifolds in the smooth category [Haefliger&Hirsch1963], and in the PL category in [Hudson1969, $\S$$\S$11], [Boechat&Haefliger1970], [Boechat1971], cf. [Vrabec1977]. The proof of [Haefliger&Hirsch1963], [Boechat&Haefliger1970], [Boechat1971] actually used the homological $k$$k$-connectedness assumption.

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].

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

By Theorem 6.2 the Whitney invariant $W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{\varepsilon(q)}$$W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{\varepsilon(q)}$ is bijective for $1\le p\le q-2$$1\le p\le q-2$. It is in fact a group isomorphism (for the group structure introduced in [Skopenkov2006a], [Skopenkov2015], [Skopenkov2015a]). The Hudson torus $\Hud(1)$$\Hud(1)$ generates $E^{p+2q+1}(S^p\times S^q)$$E^{p+2q+1}(S^p\times S^q)$ for $1\le p<q-1$$1\le p<q-1$; this holds by Theorem 6.2 because $W(\Hud(1),\Hud(0))=1\in\Zz_{\varepsilon(q)}$$W(\Hud(1),\Hud(0))=1\in\Zz_{\varepsilon(q)}$. Also, for $q\ge2$$q\ge2$ by Theorem 6.2 the Whitney invariants

$\displaystyle W^{3q}_{q-1,q}:E^{3q}_{PL}(S^{q-1}\times S^q)\to\Zz_{\varepsilon(q)} \quad\text{and}\quad W^{3q+1}_{q,q}:E^{3q+1}_{PL}(S^q\times S^q)\to\Zz_{\varepsilon(q)}\oplus \Zz_{\varepsilon(q)}$ $$\displaystyle W^{3q}_{q-1,q}:E^{3q}_{PL}(S^{q-1}\times S^q)\to\Zz_{\varepsilon(q)} \quad\text{and}\quad W^{3q+1}_{q,q}:E^{3q+1}_{PL}(S^q\times S^q)\to\Zz_{\varepsilon(q)}\oplus \Zz_{\varepsilon(q)}$$

are bijective. In the smooth category for $q$$q$ even $W^{3q}_{q-1,q}$$W^{3q}_{q-1,q}$ is not injective (see the next subsection), and $W^{3q+1}_{q,q}$$W^{3q+1}_{q,q}$ is not surjective [Boechat1971], [Skopenkov2016f], and $W^7_{2,2}$$W^7_{2,2}$ is not injective [Skopenkov2016f].

6.3 Classification in the presence of smoothly knotted spheres

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.

$\displaystyle E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)}$ $$\displaystyle E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)}$$

for each $s>1$$s>1$ [Haefliger1966]. The following result for $n=2k+3$$n=2k+3$ was obtained using the Boéchat-Haefliger formula for the smoothing obstruction [Boechat&Haefliger1970], [Boechat1971]. Using that formula one can define the higher-dimensional Kreck invariant [Skopenkov2008].

Theorem 6.3 [Skopenkov2008]. 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 each $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\}$.

E.g. by Theorem 6.3 the Whitney invariant $W:E^{6l}_D(S^{2l-1}\times S^{2l})\to\Zz$$W:E^{6l}_D(S^{2l-1}\times S^{2l})\to\Zz$ is surjective and for each $u\in\Zz$$u\in\Zz$ there is a 1-1 correspondence $W^{-1}u\to\Zz_u$$W^{-1}u\to\Zz_u$.

6.4 Classification further below the stable range

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 remarks on $E^{2n-1}(N)$$E^{2n-1}(N)$ in $\S$$\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]. 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

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

For $k=0$$k=0$ this is the same as General Position Theorem 2.1 [Skopenkov2016c] (because $V_{2n+1,n+1}$$V_{2n+1,n+1}$ is $(n-1)$$(n-1)$-connected). For $k=1$$k=1$ this is covered by Theorem 6.2; for $k\ge2$$k\ge2$ it is not.

E.g. by Theorem 6.4 there is a 1-1 correspondence $E^m(S^p\times S^q)\to\pi_p(V_{m,p+q+1})\oplus\pi_q(V_{m,p+q+1})$$E^m(S^p\times S^q)\to\pi_p(V_{m,p+q+1})\oplus\pi_q(V_{m,p+q+1})$ for $m\ge2q+3$$m\ge2q+3$ and $2m\ge3q+3p+4$$2m\ge3q+3p+4$. For a generalization see [Skopenkov2016k] (to knotted tori) and [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 each $M>n$$M>n$.

7 An orientation on the self-intersection set

Let $f:N\to\Rr^m$$f:N\to\Rr^m$ be a general position smooth map of an oriented $n$$n$-manifold $N$$N$. Assume that $m\ge n+2$$m\ge n+2$ so that the closure $Cl\Sigma(f)$$Cl\Sigma(f)$ of the self-intersection set of $f$$f$ has codimension 2 singularities.

Definition 7.1 (a natural orientation on $\Sigma(f)$$\Sigma(f)$). Take points $x,y\in N$$x,y\in N$ away from the singularities of $\Sigma(f)$$\Sigma(f)$ and such that $fx=fy$$fx=fy$. Then a $(2n-m)$$(2n-m)$-base $\xi_x$$\xi_x$ tangent to $\Sigma(f)$$\Sigma(f)$ at $x$$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)$$\Sigma(f)$ at $y$$y$. Since $N$$N$ is orientable, we can take positive $(m-n)$$(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$.

We remark that

• a change of the orientation of $N$$N$ forces changes of the signs of $\eta_x$$\eta_x$ and $\eta_y$$\eta_y$ and so does not change the orientation of $\Sigma(f)$$\Sigma(f)$.

• the natural orientation on $\Sigma(f)$$\Sigma(f)$ need not extend to $Cl\Sigma(f)$$Cl\Sigma(f)$: take the cone $D^3\to\Rr^5$$D^3\to\Rr^5$ over a general position map $S^2\to\Rr^4$$S^2\to\Rr^4$ having only one self-intersection point.

• the natural orientation on $\Sigma(f)$$\Sigma(f)$ extends to $Cl\Sigma(f)$$Cl\Sigma(f)$ if $m-n$$m-n$ is odd [Hudson1969, Lemma 11.4].

Definition 7.2 (a natural orientation on $f\Sigma(f)$$f\Sigma(f)$ for $m-n$$m-n$ even). Take a $(2n-m)$$(2n-m)$-base $\xi$$\xi$ at a point $x\in f\Sigma(f)$$x\in f\Sigma(f)$ away from the singularities of $f\Sigma(f)$$f\Sigma(f)$. Since $N$$N$ is orientable, we can take a positive $(m-n)$$(m-n)$-base $\eta_+$$\eta_+$ normal to $f\Sigma(f)$$f\Sigma(f)$ in one sheet of $f(N)$$f(N)$. Analogously construct an $(m-n)$$(m-n)$-base $\eta_-$$\eta_-$ for the other sheet of $f(N)$$f(N)$. Since $m-n$$m-n$ is even, the orientation of the base $(\xi,\eta_+,\eta_-)$$(\xi,\eta_+,\eta_-)$ of $\Rr^m$$\Rr^m$ does not depend on choosing the first and the other sheet of $f(N)$$f(N)$. 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)$$f\Sigma(f)$.

8 References

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, $\S]{Skopenkov2016c}. == Classification == ; 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, either $n$ is odd or $N$ is orientable, and either $n\ge4$ or $n=3$ and we are in the PL category. The [[#The Whitney invariant|Whitney invariant]], $$W:E^{2n}(N)\to H_1(N;\Zz_{\varepsilon(n-1)}),$$ is a 1-1 correspondence. {{endthm}} This is proved in \cite[1.3.e]{Haefliger1962b}, \cite{Haefliger1963}, \cite[Theorem 2.4]{Haefliger&Hirsch1963}, \cite[Theorem 43]{Bausum1975} in the smooth category, and in \cite{Weber1967}, \cite[Theorem 1.1]{Vrabec1977} in the PL category. If $n\ge4$ is even and $N$ is a closed connected non-orientable $n$-manifold, then \cite[Theorem 43]{Bausum1975} asserts that there is a 1-1 correspondence $$E^{2n}(N)\to \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s$$ (because $Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$ is given by multiplication with $w_1(N)$). This 1-1 correspondence can presumably be defined as a generalized Whitney invariant, but the proof used ''the Haefliger-Wu invariant'' whose definition can be found in \cite[$\S]{Skopenkov2006}. It would be interesting to check if this description of $E^{2n}(N)$ is equivalent to its different forms \cite[1.3.e]{Haefliger1962b}, \cite{Haefliger1963}, \cite[Theorem 1.1]{Vrabec1977}. The classification of ''smooth'' [[3-manifolds_in_6-space|embeddings of 3-manifolds in $\Rr^6$]] is more complicated, see $\S$\ref{s:knotted} or \cite{Skopenkov2016t}. For embeddings of $n$-manifolds in $\Rr^{2n-1}$ see [[4-manifolds_in_7-space|the case of 4-manifolds]] \cite{Skopenkov2016f}, \cite{Yasui1984} for $n\ge5$ and \cite{Saeki1999}, \cite{Skopenkov2010}, \cite{Tonkonog2010} for non-closed manifolds. Theorem \ref{th4} is generalized in $\S$\ref{s:just} to [[#Classification just below the stable range|a description]] of $E^{2n-k}(N)$ for closed $k$-connected $n$-manifolds $N$. == Hudson tori == ; Together with [[3-manifolds_in_6-space#Examples|the Haefliger knotted sphere]] \cite{Skopenkov2016t}, the examples of Hudson tori presented below were the first examples of embeddings in codimension greater than 2 which are not isotopic to the standard embedding defined below. (Hudson's construction \cite{Hudson1963} was not as explicit as those below.) For $m\ge n+2$ we define the ''standard embedding'' $S^p\times S^{n-p}\to\Rr^m$ as the composition of the standard embeddings $S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}=\Rr^{n+2}\to\Rr^m$. (Note that in \cite{Skopenkov2015}, \cite{Skopenkov2015a} a different embedding is called a standard embedding. Since these two embeddings are isotopic, no confusion would appear.) Let 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$$N \to\Rr^m$ are isotopic [Skopenkov2016c, Theorem 3.2]. In this page we summarize the situation for $m=2n\ge6$$m=2n\ge6$ and some more general situations.

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].

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, either $n$$n$ is odd or $N$$N$ is orientable, and either $n\ge4$$n\ge4$ or $n=3$$n=3$ and we are in the PL category. 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.

This is proved in [Haefliger1962b, 1.3.e], [Haefliger1963], [Haefliger&Hirsch1963, Theorem 2.4], [Bausum1975, Theorem 43] in the smooth category, and in [Weber1967], [Vrabec1977, Theorem 1.1] in the PL category.

If $n\ge4$$n\ge4$ is even and $N$$N$ is a closed connected non-orientable $n$$n$-manifold, then [Bausum1975, Theorem 43] asserts that there is a 1-1 correspondence

$\displaystyle E^{2n}(N)\to \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s$ $$\displaystyle E^{2n}(N)\to \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s$$

(because $Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$$Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$ is given by multiplication with $w_1(N)$$w_1(N)$). This 1-1 correspondence can presumably be defined as a generalized Whitney invariant, but the proof used the Haefliger-Wu invariant whose definition can be found in [Skopenkov2006, $\S$$\S$5]. It would be interesting to check if this description of $E^{2n}(N)$$E^{2n}(N)$ is equivalent to its different forms [Haefliger1962b, 1.3.e], [Haefliger1963], [Vrabec1977, Theorem 1.1].

The classification of smooth embeddings of 3-manifolds in $\Rr^6$$\Rr^6$ is more complicated, see $\S$$\S$6.3 or [Skopenkov2016t].

For embeddings of $n$$n$-manifolds in $\Rr^{2n-1}$$\Rr^{2n-1}$ see the case of 4-manifolds [Skopenkov2016f], [Yasui1984] for $n\ge5$$n\ge5$ and [Saeki1999], [Skopenkov2010], [Tonkonog2010] for non-closed manifolds.

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

3 Hudson tori

Together with the Haefliger knotted sphere [Skopenkov2016t], the examples of Hudson tori presented below were the first examples of embeddings in codimension greater than 2 which are not isotopic to the standard embedding defined below. (Hudson's construction [Hudson1963] was not as explicit as those below.)

For $m\ge n+2$$m\ge n+2$ we define the standard embedding $S^p\times S^{n-p}\to\Rr^m$$S^p\times S^{n-p}\to\Rr^m$ as the composition of the standard embeddings $S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}=\Rr^{n+2}\to\Rr^m$$S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}=\Rr^{n+2}\to\Rr^m$. (Note that in [Skopenkov2015], [Skopenkov2015a] a different embedding is called a standard embedding. Since these two embeddings are isotopic, no confusion would appear.)

Let $1_n:=(1,0,\ldots,0)\in S^n$$1_n:=(1,0,\ldots,0)\in S^n$.

Let us construct, for each $a\in\Zz$$a\in\Zz$ and $n\ge2$$n\ge2$, an embedding

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

The reader might first consider the case $n=2$$n=2$.

Definition 3.1. Take the standard embeddings $2D^{n+1}\times S^{n-1}\subset\Rr^{2n}$$2D^{n+1}\times S^{n-1}\subset\Rr^{2n}$ (where $2$$2$ means homothety with coefficient 2) and $\partial D^2\subset\partial D^{n+1}$$\partial D^2\subset\partial D^{n+1}$. Take the embedded sphere and embedded torus,

$\displaystyle 2\partial D^{n+1}\times-1_{n-1}\subset 2D^{n+1}\times S^{n-1}\subset\Rr^{2n}\quad\text{and}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$ $$\displaystyle 2\partial D^{n+1}\times-1_{n-1}\subset 2D^{n+1}\times S^{n-1}\subset\Rr^{2n}\quad\text{and}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$

Join them by an arc whose interior misses the two embedded manifolds. The Hudson torus $\Hud_n(1)$$\Hud_n(1)$ is the embedded connected sum of the two embeddings along this arc, compatible with the orientation. (Unlike the unlinked embedded connected sum [Skopenkov2016c] this is a linked embedded connected sum, i.e. the connected sum of two embeddings whose images are not contained in disjoint cubes.)

We remark that the construction in Definition 3.1 works for $n=1$$n=1$. This is not the case for the next construction in Definition 3.2.

Definition 3.2. For $a\in\Zz$$a\in\Zz$ we repeat the construction of Definition 3.1 replacing $2\partial D^{n+1}\times-1_{n-1}$$2\partial D^{n+1}\times-1_{n-1}$ by $|a|$$|a|$ copies $(1+\frac1k)\partial D^{n+1}\times-1_{n-1}$$(1+\frac1k)\partial D^{n+1}\times-1_{n-1}$ ($k=1,\dots,|a|$$k=1,\dots,|a|$) of $S^n$$S^n$. The copies are outside $D^{n+1}\times S^{n-1}$$D^{n+1}\times S^{n-1}$ and are `parallel' to $\partial D^{n+1}\times-1_{n-1}$$\partial D^{n+1}\times-1_{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 by tubes joining each $k$$k$-th copy to the $(k+1)$$(k+1)$-th copy. We obtain an embedding $g:S^n\to\Rr^{2n}$$g:S^n\to\Rr^{2n}$. Let $\Hud_n(a)$$\Hud_n(a)$ be the linked embedded connected sum of $g$$g$ with the embedding $\partial D^2\times S^{n-1}\subset\Rr^{2n}$$\partial D^2\times S^{n-1}\subset\Rr^{2n}$ from Definition 3.1.

Clearly, $\Hud_n(0)$$\Hud_n(0)$ is isotopic to the standard embedding.

The original motivation for Hudson was that $\Hud_n(1)$$\Hud_n(1)$ is not isotopic to $\Hud_n(0)$$\Hud_n(0)$ for each $n$$n$ (this is a particular case of Proposition 3.3 below).

One guesses 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 exists and is `realized' by the homotopy class of the map

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

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

Proposition 3.3. 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 3.3 follows by calculation of the Whitney invariant (Remark 5.3.e below) and, for $n$$n$ even, by Theorem 2.1.

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)$ isotopic to $\Hud_2(2)$$\Hud_2(2)$? It would also be interesting to find an explicit construction of an isotopy between $\Hud_{2k}(a)$$\Hud_{2k}(a)$ and $\Hud_{2k}(a+2)$$\Hud_{2k}(a+2)$, cf. [Vrabec1977, $\S$$\S$5].

Definition 3.4. Let us give, for $a\in\Zz$$a\in\Zz$ and $n\ge2$$n\ge2$, another construction of embeddings

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

Define a map $S^0\times S^{n-1}\to D^n$$S^0\times S^{n-1}\to D^n$ to be the constant $0\in D^n$$0\in D^n$ on one component $1\times S^{n-1}$$1\times S^{n-1}$ and the `standard inclusion' $\{-1\}\times S^{n-1}\to\partial D^n\subset D^n$$\{-1\}\times S^{n-1}\to\partial D^n\subset D^n$ on the other component. This map gives an embedding

$\displaystyle S^0\times S^{n-1}\to D^n\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$ $$\displaystyle S^0\times S^{n-1}\to D^n\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$

(See [Skopenkov2006, Figure 2.2]. The image of this embedding is the union of the standard $S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$$S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$ and the graph of the identity map in $S^{n-1}\times S^{n-1}\subset D^n\times S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$$S^{n-1}\times S^{n-1}\subset D^n\times S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$.)

Take any $x\in S^{n-1}$$x\in S^{n-1}$. The disk $D^{n+1}\times x$$D^{n+1}\times x$ intersects the image of this embedding at two points lying in $D^n\times x$$D^n\times x$, i.e., at the image of an embedding $S^0\times x\to D^n\times x$$S^0\times x\to D^n\times x$. Extend the latter embedding to an embedding $S^1\times x\to D^{n+1}\times x$$S^1\times x\to D^{n+1}\times x$. See [Skopenkov2006, Figure 2.3].) Thus we obtain the Hudson torus

$\displaystyle \Hud_n'(1):S^1\times S^{n-1}\overset h\to D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$ $$\displaystyle \Hud_n'(1):S^1\times S^{n-1}\overset h\to D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$

Here $h(e^{i\theta},y):=(y\cos\theta,\sin\theta,y)$$h(e^{i\theta},y):=(y\cos\theta,\sin\theta,y)$, where $D^{n+1}$$D^{n+1}$ is identified with $D^n\times D^1$$D^n\times D^1$.

The embedding $\Hud_n'(a)$$\Hud_n'(a)$ is obtained in the same way starting from a map $\varphi:\{-1\}\times S^{n-1}\to\partial D^n$$\varphi:\{-1\}\times S^{n-1}\to\partial D^n$ of degree $a$$a$ instead of the `standard inclusion'.

Remark 3.5. (a) The analogue of Proposition 3.3 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$$n\ge4$ and are PL isotopic for $n\ge3$$n\ge3$ [Skopenkov2006a]. It would be interesting to know if they are isotopic for $n=2$$n=2$, or if they are smoothly isotopic for $n=3$$n=3$.

(c) For $n=2$$n=2$ these construction give what we call the left Hudson torus. The right Hudson torus is constructed analogously. It is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of $S^1\times S^1$$S^1\times S^1$. The right and the left Hudson tori are not isotopic by Remark 5.3.e 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$ 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})$. There are versions of these constructions corresponding to Definition 3.4. For $p=0$$p=0$ this corresponds to the Zeeman construction [Skopenkov2016h] and its composition with the second unframed Kirby move. It would be interesting to know if the links $\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$$\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$ are isotopic, cf. [Skopenkov2015a, Remark 2.9.b]. These constructions can be further generalized [Skopenkov2016k].

4 Action by linked embedded connected sum

In this section we generalize the construction of the Hudson torus $\Hud(a)$$\Hud(a)$. For $n\ge3$$n\ge3$, a closed connected orientable $n$$n$-manifold $N$$N$, an embedding $f_0:N\to\Rr^{2n}$$f_0:N\to\Rr^{2n}$ and $a\in H_1(N;\Zz_{\varepsilon(n-1)})$$a\in H_1(N;\Zz_{\varepsilon(n-1)})$, we construct an embedding $f_a:N\to\Rr^{2n}$$f_a:N\to\Rr^{2n}$. This embedding is obtained by linked embedded connected sum of $f_0$$f_0$ with an $n$$n$-sphere representing the homology Alexander dual of $a$$a$.

More precisely, represent $a$$a$ by an embedding $a:S^1\to N$$a:S^1\to N$. 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}$. Identify $\nu_{f_0}^{-1}a(S^1)$$\nu_{f_0}^{-1}a(S^1)$ with $S^1\times S^{n-1}$$S^1\times S^{n-1}$. In the next paragraph we recall definition of embedded surgery on $S^1\times*\subset S^1\times S^{n-1}$$S^1\times*\subset S^1\times S^{n-1}$, which yields an embedding $g:S^n\to C_{f_0}$$g:S^n\to C_{f_0}$. Then we define $f_a$$f_a$ to be the (linked) embedded connected sum of $f_0$$f_0$ and $g$$g$ (along a certain arc joining their images).

Take a vector field on $S^1\times-1_{n-1}$$S^1\times-1_{n-1}$ normal to $S^1\times S^{n-1}$$S^1\times S^{n-1}$. Extend $S^1\times-1_{n-1}$$S^1\times-1_{n-1}$ along this vector field to a map $\overline a:D^2\to\Rr^{2n}$$\overline a:D^2\to\Rr^{2n}$. Since $2n>4$$2n>4$ and $n+2<2n$$n+2<2n$, by general position we may assume that $\overline a$$\overline a$ is an embedding and $\overline a(Int D^2)$$\overline a(Int D^2)$ misses $f_0(N)\cup S^1\times S^{n-1}$$f_0(N)\cup S^1\times S^{n-1}$. Since $n-1>1$$n-1>1$, we have $\pi_1(V_{2n-2,n-1})=0$$\pi_1(V_{2n-2,n-1})=0$. Hence the standard framing of $S^1\times-1_{n-1}$$S^1\times-1_{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 a(D^2)$$\overline a(D^2)$ in $\Rr^{2n}$$\Rr^{2n}$. Thus $\overline a$$\overline a$ extends to an embedding

$\displaystyle \widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}.$ $$\displaystyle \widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}.$$

Define an embedding $g:S^n\to C_{f_0}$$g:S^n\to C_{f_0}$ by setting

$\displaystyle g(S^n):\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n$ $$\displaystyle g(S^n):\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n$$

with natural orientation.

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

Proposition 4.1. Linked embedded connected sum, or parametric connected sum, 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)$, unless $n=3$$n=3$ in the smooth category.

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

5 The Whitney invariant

Let $N$$N$ be a closed $n$$n$-manifold and fix an embedding $f_0:N\to\Rr^m$$f_0:N\to\Rr^m$. For any other embedding $f \colon N \to \Rr^m$$f \colon N \to \Rr^m$, there is an invariant $W(f, f_0)$$W(f, f_0)$, called the Whitney invariant, which we define in this section. Roughly speaking, $W(f,f_0)=W_{f_0}(f)=W(f)$$W(f,f_0)=W_{f_0}(f)=W(f)$ is defined as the homology class of the self-intersection set $\Sigma(H)$$\Sigma(H)$ of a general position homotopy $H$$H$ between $f$$f$ and $f_0$$f_0$. This is formalized in Definition 5.2 in the smooth category, following [Skopenkov2010]. The definition in the PL category is analogous [Hudson1969, $\S$$\S$12], [Vrabec1977, p. 145], [Skopenkov2006, $\S$$\S$2.4]. We begin by presenting a simpler definition of the Whitney invariant, Definition 5.1, for a particular case. For Theorem 2.1 only the case $m=2n$$m=2n$ is required.

Fix an orientation on $\Rr^m$$\Rr^m$. Assume that either $m-n$$m-n$ is even or $N$$N$ is oriented.

Definition 5.1. Assume that $N$$N$ is $(2n-m)$$(2n-m)$-connected and $2m\ge3n+3$$2m\ge3n+3$. Then restrictions of $f$$f$ and $f_0$$f_0$ to $N_0$$N_0$ are regular homotopic [Hirsch1959]. 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, cf. [Haefliger&Hirsch1963, 3.1.b], [Takase2006, Lemma 2.2]. 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$$n+2(n+1)<2m$, by general position $Cl(f\cap F)$$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. For $m-n$$m-n$ odd it has a natural orientation defined below, and so carries a homology class with $\Zz$$\Zz$ coefficients. Define $W(f)$$W(f)$ to be the homology class:

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

The orientation on $f\cap F$$f\cap F$ (extendable to $Cl(f\cap F)$$Cl(f\cap F)$) is defined for $m-n$$m-n$ odd as follows. For each point $x\in f\cap F$$x\in f\cap F$ take a vector at $x$$x$ tangent to $f\cap F$$f\cap F$. Complete this vector to a positive base tangent to $N$$N$. Since $n+2(n+1)<2m$$n+2(n+1)<2m$, by general position there is a unique point $y\in B^n\times I$$y\in B^n\times I$ such that $Fy=fx$$Fy=fx$. The tangent vector at $x$$x$ thus gives a tangent vector at $y$$y$ to $B^n\times I$$B^n\times I$. Complete this vector to a positive base tangent to $B^n\times I$$B^n\times I$, where the orientation on $B^n$$B^n$ comes from $N$$N$. The union of the images of the constructed two bases is a base at $Fy=fx$$Fy=fx$ of $\Rr^m$$\Rr^m$. If the latter base is positive, then call the initial vector 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 the latter base of $\Rr^m$$\Rr^m$, this condition indeed defines an orientation on $f\cap F$$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$. Since $m\ge n+2$$m\ge n+2$, by general position the closure $Cl\Sigma(H)$$Cl\Sigma(H)$ of the self-intersection set has codimension 2 singularities. So the closure carries a homology class with $\Zz_2$$\Zz_2$ coefficients. (Note that $Cl\Sigma(H)$$Cl\Sigma(H)$ can be assumed to be a submanifold for $2m\ge3n+2$$2m\ge3n+2$.) For $m-n$$m-n$ odd it has a natural orientation ($\S$$\S$7) and so carries a homology class with $\Zz$$\Zz$ coefficients. Define the Whitney invariant to be the homology class:

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

(The orientation is defined for each $m,n$$m,n$ but used only for odd $m-n$$m-n$. When $m-n$$m-n$ is even, for $W(f)$$W(f)$ being well-defined we need $\Zz_2$$\Zz_2$-coefficients.)

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

$\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_{\varepsilon(m-n-1)}),\quad [f] \mapsto W(f) = W(f, f_0).$ $$\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_{\varepsilon(m-n-1)}),\quad [f] \mapsto W(f) = W(f, f_0).$$

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 $H$$H$, which is proved analogously to [Skopenkov2006, $\S$$\S$2.4].

(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 $F$$F$ to be fixed on $N_0$$N_0$.) 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) Since a change of the orientation on $N$$N$ forces a change of the orientation on $B^n$$B^n$, the class $W(f)$$W(f)$ is independent of the choice of the orientation on $N$$N$. 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 we may assume that $f=f_0=\sigma\circ f$$f=f_0=\sigma\circ f$ on $N_0$$N_0$ and because a change of the orientation of $\Rr^m$$\Rr^m$ forces a change of the orientation of $f\cap F$$f\cap F$).

(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$, is $(a\mod2,0)$$(a\mod2,0)$ for $n=2$$n=2$.

(e) $W(f\#g)=W(f)$$W(f\#g)=W(f)$ for each pair of embeddings $f:N\to\Rr^m$$f:N\to\Rr^m$ and $g:S^n\to\Rr^m$$g:S^n\to\Rr^m$.

(f) For $m=2n+1$$m=2n+1$ the Whitney invariant can be recovered from the collection of pairwise linking coefficients of the components of $N$$N$, cf. Remark 3.2.b of [Skopenkov2016h].

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] 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 (Linked embedded connected sum, cf. [Skopenkov2010, Definition 1.4]). If $N$$N$ is $k$$k$-connected, then for an embedding $f_0:N\to S^{2n-k}$$f_0:N\to S^{2n-k}$ and a class $a\in H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$$a\in H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$ one can construct an embedding $f_a:N\to S^{2n-k}$$f_a:N\to S^{2n-k}$ by linked connected sum analogously to the case $k=0$$k=0$.

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$$n\ge2k+4$ in the PL or smooth categories, respectively). If $n=2k+3$$n=2k+3$, then this construction gives only a construction of embeddings $f_a:N\to\Rr^{2n-k}$$f_a:N\to\Rr^{2n-k}$ for each $a\in H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$$a\in H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$ but not a well-defined action of $H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$$H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$ on $E^{2n-k}(N)$$E^{2n-k}(N)$.

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

6.2 Classification just below the stable range

Theorem 6.2. Let $N$$N$ be a closed orientable homologically $k$$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\ge k+3$$n\ge k+3$ in the smooth category or $n\ge2k+4$$n\ge2k+4$ in the categoriy.

This was proved for $k$$k$-connected manifolds in the smooth category [Haefliger&Hirsch1963], and in the PL category in [Hudson1969, $\S$$\S$11], [Boechat&Haefliger1970], [Boechat1971], cf. [Vrabec1977]. The proof of [Haefliger&Hirsch1963], [Boechat&Haefliger1970], [Boechat1971] actually used the homological $k$$k$-connectedness assumption.

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].

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

By Theorem 6.2 the Whitney invariant $W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{\varepsilon(q)}$$W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{\varepsilon(q)}$ is bijective for $1\le p\le q-2$$1\le p\le q-2$. It is in fact a group isomorphism (for the group structure introduced in [Skopenkov2006a], [Skopenkov2015], [Skopenkov2015a]). The Hudson torus $\Hud(1)$$\Hud(1)$ generates $E^{p+2q+1}(S^p\times S^q)$$E^{p+2q+1}(S^p\times S^q)$ for $1\le p<q-1$$1\le p<q-1$; this holds by Theorem 6.2 because $W(\Hud(1),\Hud(0))=1\in\Zz_{\varepsilon(q)}$$W(\Hud(1),\Hud(0))=1\in\Zz_{\varepsilon(q)}$. Also, for $q\ge2$$q\ge2$ by Theorem 6.2 the Whitney invariants

$\displaystyle W^{3q}_{q-1,q}:E^{3q}_{PL}(S^{q-1}\times S^q)\to\Zz_{\varepsilon(q)} \quad\text{and}\quad W^{3q+1}_{q,q}:E^{3q+1}_{PL}(S^q\times S^q)\to\Zz_{\varepsilon(q)}\oplus \Zz_{\varepsilon(q)}$ $$\displaystyle W^{3q}_{q-1,q}:E^{3q}_{PL}(S^{q-1}\times S^q)\to\Zz_{\varepsilon(q)} \quad\text{and}\quad W^{3q+1}_{q,q}:E^{3q+1}_{PL}(S^q\times S^q)\to\Zz_{\varepsilon(q)}\oplus \Zz_{\varepsilon(q)}$$

are bijective. In the smooth category for $q$$q$ even $W^{3q}_{q-1,q}$$W^{3q}_{q-1,q}$ is not injective (see the next subsection), and $W^{3q+1}_{q,q}$$W^{3q+1}_{q,q}$ is not surjective [Boechat1971], [Skopenkov2016f], and $W^7_{2,2}$$W^7_{2,2}$ is not injective [Skopenkov2016f].

6.3 Classification in the presence of smoothly knotted spheres

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.

$\displaystyle E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)}$ $$\displaystyle E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)}$$

for each $s>1$$s>1$ [Haefliger1966]. The following result for $n=2k+3$$n=2k+3$ was obtained using the Boéchat-Haefliger formula for the smoothing obstruction [Boechat&Haefliger1970], [Boechat1971]. Using that formula one can define the higher-dimensional Kreck invariant [Skopenkov2008].

Theorem 6.3 [Skopenkov2008]. 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 each $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\}$.

E.g. by Theorem 6.3 the Whitney invariant $W:E^{6l}_D(S^{2l-1}\times S^{2l})\to\Zz$$W:E^{6l}_D(S^{2l-1}\times S^{2l})\to\Zz$ is surjective and for each $u\in\Zz$$u\in\Zz$ there is a 1-1 correspondence $W^{-1}u\to\Zz_u$$W^{-1}u\to\Zz_u$.

6.4 Classification further below the stable range

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 remarks on $E^{2n-1}(N)$$E^{2n-1}(N)$ in $\S$$\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]. 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

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

For $k=0$$k=0$ this is the same as General Position Theorem 2.1 [Skopenkov2016c] (because $V_{2n+1,n+1}$$V_{2n+1,n+1}$ is $(n-1)$$(n-1)$-connected). For $k=1$$k=1$ this is covered by Theorem 6.2; for $k\ge2$$k\ge2$ it is not.

E.g. by Theorem 6.4 there is a 1-1 correspondence $E^m(S^p\times S^q)\to\pi_p(V_{m,p+q+1})\oplus\pi_q(V_{m,p+q+1})$$E^m(S^p\times S^q)\to\pi_p(V_{m,p+q+1})\oplus\pi_q(V_{m,p+q+1})$ for $m\ge2q+3$$m\ge2q+3$ and $2m\ge3q+3p+4$$2m\ge3q+3p+4$. For a generalization see [Skopenkov2016k] (to knotted tori) and [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 each $M>n$$M>n$.

7 An orientation on the self-intersection set

Let $f:N\to\Rr^m$$f:N\to\Rr^m$ be a general position smooth map of an oriented $n$$n$-manifold $N$$N$. Assume that $m\ge n+2$$m\ge n+2$ so that the closure $Cl\Sigma(f)$$Cl\Sigma(f)$ of the self-intersection set of $f$$f$ has codimension 2 singularities.

Definition 7.1 (a natural orientation on $\Sigma(f)$$\Sigma(f)$). Take points $x,y\in N$$x,y\in N$ away from the singularities of $\Sigma(f)$$\Sigma(f)$ and such that $fx=fy$$fx=fy$. Then a $(2n-m)$$(2n-m)$-base $\xi_x$$\xi_x$ tangent to $\Sigma(f)$$\Sigma(f)$ at $x$$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)$$\Sigma(f)$ at $y$$y$. Since $N$$N$ is orientable, we can take positive $(m-n)$$(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$.

We remark that

• a change of the orientation of $N$$N$ forces changes of the signs of $\eta_x$$\eta_x$ and $\eta_y$$\eta_y$ and so does not change the orientation of $\Sigma(f)$$\Sigma(f)$.

• the natural orientation on $\Sigma(f)$$\Sigma(f)$ need not extend to $Cl\Sigma(f)$$Cl\Sigma(f)$: take the cone $D^3\to\Rr^5$$D^3\to\Rr^5$ over a general position map $S^2\to\Rr^4$$S^2\to\Rr^4$ having only one self-intersection point.

• the natural orientation on $\Sigma(f)$$\Sigma(f)$ extends to $Cl\Sigma(f)$$Cl\Sigma(f)$ if $m-n$$m-n$ is odd [Hudson1969, Lemma 11.4].

Definition 7.2 (a natural orientation on $f\Sigma(f)$$f\Sigma(f)$ for $m-n$$m-n$ even). Take a $(2n-m)$$(2n-m)$-base $\xi$$\xi$ at a point $x\in f\Sigma(f)$$x\in f\Sigma(f)$ away from the singularities of $f\Sigma(f)$$f\Sigma(f)$. Since $N$$N$ is orientable, we can take a positive $(m-n)$$(m-n)$-base $\eta_+$$\eta_+$ normal to $f\Sigma(f)$$f\Sigma(f)$ in one sheet of $f(N)$$f(N)$. Analogously construct an $(m-n)$$(m-n)$-base $\eta_-$$\eta_-$ for the other sheet of $f(N)$$f(N)$. Since $m-n$$m-n$ is even, the orientation of the base $(\xi,\eta_+,\eta_-)$$(\xi,\eta_+,\eta_-)$ of $\Rr^m$$\Rr^m$ does not depend on choosing the first and the other sheet of $f(N)$$f(N)$. 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)$$f\Sigma(f)$.

8 References

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_n:=(1,0,\ldots,0)\in S^n$. Let us construct, for each $a\in\Zz$ and $n\ge2$, an embedding $$\Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.$$ The reader might first consider the case $n=2$. {{beginthm|Definition}}\label{dh11} Take the standard embeddings D^{n+1}\times S^{n-1}\subset\Rr^{2n}$ (where $ means [[Wikipedia:Homothetic_transformation|homothety]] with coefficient 2) and $\partial D^2\subset\partial D^{n+1}$. Take the embedded sphere and embedded torus, $\partial D^{n+1}\times-1_{n-1}\subset 2D^{n+1}\times S^{n-1}\subset\Rr^{2n}\quad\text{and}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$ Join them by an arc whose interior misses the two embedded manifolds. ''The Hudson torus'' $\Hud_n(1)$ is the embedded connected sum of the two embeddings along this arc, compatible with the orientation. (Unlike the [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Embedded connected sum|unlinked embedded connected sum]] \cite{Skopenkov2016c} this is a ''linked'' embedded connected sum, i.e. the connected sum of two embeddings whose images are ''not'' contained in disjoint cubes.) {{endthm}} We remark that the construction in Definition \ref{dh11} works for $n=1$. This is not the case for the next construction in Definition \ref{dh1n}. {{beginthm|Definition}}\label{dh1n} For $a\in\Zz$ we repeat the construction of Definition \ref{dh11} replacing \partial D^{n+1}\times-1_{n-1}$ by $|a|$ copies $(1+\frac1k)\partial D^{n+1}\times-1_{n-1}$ ($k=1,\dots,|a|$) of $S^n$. The copies are outside $D^{n+1}\times S^{n-1}$ and are `parallel' to $\partial D^{n+1}\times-1_{n-1}$. The copies have the standard orientation for $a>0$ or the opposite orientation for $a<0$. Then we make embedded connected sum by tubes joining each $k$-th copy to the $(k+1)$-th copy. We obtain an embedding $g:S^n\to\Rr^{2n}$. Let $\Hud_n(a)$ be the linked embedded connected sum of $g$ with the embedding $\partial D^2\times S^{n-1}\subset\Rr^{2n}$ from Definition \ref{dh11}. {{endthm}} Clearly, $\Hud_n(0)$ is isotopic to the standard embedding. The original motivation for Hudson was that $\Hud_n(1)$ is not isotopic to $\Hud_n(0)$ for each $n$ (this is a particular case of Proposition \ref{pr3} below). One guesses that $\Hud_n(a)$ is not isotopic to $\Hud_n(a')$ for $a\ne a'$. And that a $\Zz$-valued invariant exists and is `realized' by the homotopy class of the map $$S^n\overset g\to S^{2n}-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.$$ However, this is only true for $n$ odd. {{beginthm|Proposition}}\label{pr3} For $n\ge3$ odd $\Hud_n(a)$ is isotopic to $\Hud_n(a')$ if and only if $a=a'$. For $n\ge4$ even $\Hud_n(a)$ is isotopic to $\Hud_n(a')$ if and only if $a\equiv a'\text{mod } 2$. {{endthm}} Proposition \ref{pr3} follows by calculation of [[#The_Whitney_invariant|the Whitney invariant]] (Remark \ref{re5}.e below) and, for $n$ even, by Theorem \ref{th4}. Analogously, $\Hud_2(a)$ is not isotopic to $\Hud_2(a')$ if $a\not\equiv a' \text{mod } 2$. It would be interesting to know if the converse holds, e.g. is $\Hud_2(0)$ isotopic to $\Hud_2(2)$? It would also be interesting to find an explicit construction of an isotopy between $\Hud_{2k}(a)$ and $\Hud_{2k}(a+2)$, cf. \cite[$\S]{Vrabec1977}. {{beginthm|Definition}}\label{dh2n} Let us give, for $a\in\Zz$ and $n\ge2$, another construction of embeddings $$\Hud_n'(a):S^1\times S^{n-1}\to\Rr^{2n}.$$ Define a map $S^0\times S^{n-1}\to D^n$ to be the constant $N$ is a connected manifold of dimension $n>1$$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]. In this page we summarize the situation for $m=2n\ge6$$m=2n\ge6$ and some more general situations.

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].

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, either $n$$n$ is odd or $N$$N$ is orientable, and either $n\ge4$$n\ge4$ or $n=3$$n=3$ and we are in the PL category. 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.

This is proved in [Haefliger1962b, 1.3.e], [Haefliger1963], [Haefliger&Hirsch1963, Theorem 2.4], [Bausum1975, Theorem 43] in the smooth category, and in [Weber1967], [Vrabec1977, Theorem 1.1] in the PL category.

If $n\ge4$$n\ge4$ is even and $N$$N$ is a closed connected non-orientable $n$$n$-manifold, then [Bausum1975, Theorem 43] asserts that there is a 1-1 correspondence

$\displaystyle E^{2n}(N)\to \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s$ $$\displaystyle E^{2n}(N)\to \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s$$

(because $Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$$Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$ is given by multiplication with $w_1(N)$$w_1(N)$). This 1-1 correspondence can presumably be defined as a generalized Whitney invariant, but the proof used the Haefliger-Wu invariant whose definition can be found in [Skopenkov2006, $\S$$\S$5]. It would be interesting to check if this description of $E^{2n}(N)$$E^{2n}(N)$ is equivalent to its different forms [Haefliger1962b, 1.3.e], [Haefliger1963], [Vrabec1977, Theorem 1.1].

The classification of smooth embeddings of 3-manifolds in $\Rr^6$$\Rr^6$ is more complicated, see $\S$$\S$6.3 or [Skopenkov2016t].

For embeddings of $n$$n$-manifolds in $\Rr^{2n-1}$$\Rr^{2n-1}$ see the case of 4-manifolds [Skopenkov2016f], [Yasui1984] for $n\ge5$$n\ge5$ and [Saeki1999], [Skopenkov2010], [Tonkonog2010] for non-closed manifolds.

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

3 Hudson tori

Together with the Haefliger knotted sphere [Skopenkov2016t], the examples of Hudson tori presented below were the first examples of embeddings in codimension greater than 2 which are not isotopic to the standard embedding defined below. (Hudson's construction [Hudson1963] was not as explicit as those below.)

For $m\ge n+2$$m\ge n+2$ we define the standard embedding $S^p\times S^{n-p}\to\Rr^m$$S^p\times S^{n-p}\to\Rr^m$ as the composition of the standard embeddings $S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}=\Rr^{n+2}\to\Rr^m$$S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}=\Rr^{n+2}\to\Rr^m$. (Note that in [Skopenkov2015], [Skopenkov2015a] a different embedding is called a standard embedding. Since these two embeddings are isotopic, no confusion would appear.)

Let $1_n:=(1,0,\ldots,0)\in S^n$$1_n:=(1,0,\ldots,0)\in S^n$.

Let us construct, for each $a\in\Zz$$a\in\Zz$ and $n\ge2$$n\ge2$, an embedding

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

The reader might first consider the case $n=2$$n=2$.

Definition 3.1. Take the standard embeddings $2D^{n+1}\times S^{n-1}\subset\Rr^{2n}$$2D^{n+1}\times S^{n-1}\subset\Rr^{2n}$ (where $2$$2$ means homothety with coefficient 2) and $\partial D^2\subset\partial D^{n+1}$$\partial D^2\subset\partial D^{n+1}$. Take the embedded sphere and embedded torus,

$\displaystyle 2\partial D^{n+1}\times-1_{n-1}\subset 2D^{n+1}\times S^{n-1}\subset\Rr^{2n}\quad\text{and}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$ $$\displaystyle 2\partial D^{n+1}\times-1_{n-1}\subset 2D^{n+1}\times S^{n-1}\subset\Rr^{2n}\quad\text{and}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$

Join them by an arc whose interior misses the two embedded manifolds. The Hudson torus $\Hud_n(1)$$\Hud_n(1)$ is the embedded connected sum of the two embeddings along this arc, compatible with the orientation. (Unlike the unlinked embedded connected sum [Skopenkov2016c] this is a linked embedded connected sum, i.e. the connected sum of two embeddings whose images are not contained in disjoint cubes.)

We remark that the construction in Definition 3.1 works for $n=1$$n=1$. This is not the case for the next construction in Definition 3.2.

Definition 3.2. For $a\in\Zz$$a\in\Zz$ we repeat the construction of Definition 3.1 replacing $2\partial D^{n+1}\times-1_{n-1}$$2\partial D^{n+1}\times-1_{n-1}$ by $|a|$$|a|$ copies $(1+\frac1k)\partial D^{n+1}\times-1_{n-1}$$(1+\frac1k)\partial D^{n+1}\times-1_{n-1}$ ($k=1,\dots,|a|$$k=1,\dots,|a|$) of $S^n$$S^n$. The copies are outside $D^{n+1}\times S^{n-1}$$D^{n+1}\times S^{n-1}$ and are `parallel' to $\partial D^{n+1}\times-1_{n-1}$$\partial D^{n+1}\times-1_{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 by tubes joining each $k$$k$-th copy to the $(k+1)$$(k+1)$-th copy. We obtain an embedding $g:S^n\to\Rr^{2n}$$g:S^n\to\Rr^{2n}$. Let $\Hud_n(a)$$\Hud_n(a)$ be the linked embedded connected sum of $g$$g$ with the embedding $\partial D^2\times S^{n-1}\subset\Rr^{2n}$$\partial D^2\times S^{n-1}\subset\Rr^{2n}$ from Definition 3.1.

Clearly, $\Hud_n(0)$$\Hud_n(0)$ is isotopic to the standard embedding.

The original motivation for Hudson was that $\Hud_n(1)$$\Hud_n(1)$ is not isotopic to $\Hud_n(0)$$\Hud_n(0)$ for each $n$$n$ (this is a particular case of Proposition 3.3 below).

One guesses 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 exists and is `realized' by the homotopy class of the map

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

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

Proposition 3.3. 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 3.3 follows by calculation of the Whitney invariant (Remark 5.3.e below) and, for $n$$n$ even, by Theorem 2.1.

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)$ isotopic to $\Hud_2(2)$$\Hud_2(2)$? It would also be interesting to find an explicit construction of an isotopy between $\Hud_{2k}(a)$$\Hud_{2k}(a)$ and $\Hud_{2k}(a+2)$$\Hud_{2k}(a+2)$, cf. [Vrabec1977, $\S$$\S$5].

Definition 3.4. Let us give, for $a\in\Zz$$a\in\Zz$ and $n\ge2$$n\ge2$, another construction of embeddings

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

Define a map $S^0\times S^{n-1}\to D^n$$S^0\times S^{n-1}\to D^n$ to be the constant $0\in D^n$$0\in D^n$ on one component $1\times S^{n-1}$$1\times S^{n-1}$ and the `standard inclusion' $\{-1\}\times S^{n-1}\to\partial D^n\subset D^n$$\{-1\}\times S^{n-1}\to\partial D^n\subset D^n$ on the other component. This map gives an embedding

$\displaystyle S^0\times S^{n-1}\to D^n\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$ $$\displaystyle S^0\times S^{n-1}\to D^n\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$

(See [Skopenkov2006, Figure 2.2]. The image of this embedding is the union of the standard $S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$$S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$ and the graph of the identity map in $S^{n-1}\times S^{n-1}\subset D^n\times S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$$S^{n-1}\times S^{n-1}\subset D^n\times S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$.)

Take any $x\in S^{n-1}$$x\in S^{n-1}$. The disk $D^{n+1}\times x$$D^{n+1}\times x$ intersects the image of this embedding at two points lying in $D^n\times x$$D^n\times x$, i.e., at the image of an embedding $S^0\times x\to D^n\times x$$S^0\times x\to D^n\times x$. Extend the latter embedding to an embedding $S^1\times x\to D^{n+1}\times x$$S^1\times x\to D^{n+1}\times x$. See [Skopenkov2006, Figure 2.3].) Thus we obtain the Hudson torus

$\displaystyle \Hud_n'(1):S^1\times S^{n-1}\overset h\to D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$ $$\displaystyle \Hud_n'(1):S^1\times S^{n-1}\overset h\to D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$

Here $h(e^{i\theta},y):=(y\cos\theta,\sin\theta,y)$$h(e^{i\theta},y):=(y\cos\theta,\sin\theta,y)$, where $D^{n+1}$$D^{n+1}$ is identified with $D^n\times D^1$$D^n\times D^1$.

The embedding $\Hud_n'(a)$$\Hud_n'(a)$ is obtained in the same way starting from a map $\varphi:\{-1\}\times S^{n-1}\to\partial D^n$$\varphi:\{-1\}\times S^{n-1}\to\partial D^n$ of degree $a$$a$ instead of the `standard inclusion'.

Remark 3.5. (a) The analogue of Proposition 3.3 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$$n\ge4$ and are PL isotopic for $n\ge3$$n\ge3$ [Skopenkov2006a]. It would be interesting to know if they are isotopic for $n=2$$n=2$, or if they are smoothly isotopic for $n=3$$n=3$.

(c) For $n=2$$n=2$ these construction give what we call the left Hudson torus. The right Hudson torus is constructed analogously. It is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of $S^1\times S^1$$S^1\times S^1$. The right and the left Hudson tori are not isotopic by Remark 5.3.e 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$ 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})$. There are versions of these constructions corresponding to Definition 3.4. For $p=0$$p=0$ this corresponds to the Zeeman construction [Skopenkov2016h] and its composition with the second unframed Kirby move. It would be interesting to know if the links $\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$$\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$ are isotopic, cf. [Skopenkov2015a, Remark 2.9.b]. These constructions can be further generalized [Skopenkov2016k].

4 Action by linked embedded connected sum

In this section we generalize the construction of the Hudson torus $\Hud(a)$$\Hud(a)$. For $n\ge3$$n\ge3$, a closed connected orientable $n$$n$-manifold $N$$N$, an embedding $f_0:N\to\Rr^{2n}$$f_0:N\to\Rr^{2n}$ and $a\in H_1(N;\Zz_{\varepsilon(n-1)})$$a\in H_1(N;\Zz_{\varepsilon(n-1)})$, we construct an embedding $f_a:N\to\Rr^{2n}$$f_a:N\to\Rr^{2n}$. This embedding is obtained by linked embedded connected sum of $f_0$$f_0$ with an $n$$n$-sphere representing the homology Alexander dual of $a$$a$.

More precisely, represent $a$$a$ by an embedding $a:S^1\to N$$a:S^1\to N$. 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}$. Identify $\nu_{f_0}^{-1}a(S^1)$$\nu_{f_0}^{-1}a(S^1)$ with $S^1\times S^{n-1}$$S^1\times S^{n-1}$. In the next paragraph we recall definition of embedded surgery on $S^1\times*\subset S^1\times S^{n-1}$$S^1\times*\subset S^1\times S^{n-1}$, which yields an embedding $g:S^n\to C_{f_0}$$g:S^n\to C_{f_0}$. Then we define $f_a$$f_a$ to be the (linked) embedded connected sum of $f_0$$f_0$ and $g$$g$ (along a certain arc joining their images).

Take a vector field on $S^1\times-1_{n-1}$$S^1\times-1_{n-1}$ normal to $S^1\times S^{n-1}$$S^1\times S^{n-1}$. Extend $S^1\times-1_{n-1}$$S^1\times-1_{n-1}$ along this vector field to a map $\overline a:D^2\to\Rr^{2n}$$\overline a:D^2\to\Rr^{2n}$. Since $2n>4$$2n>4$ and $n+2<2n$$n+2<2n$, by general position we may assume that $\overline a$$\overline a$ is an embedding and $\overline a(Int D^2)$$\overline a(Int D^2)$ misses $f_0(N)\cup S^1\times S^{n-1}$$f_0(N)\cup S^1\times S^{n-1}$. Since $n-1>1$$n-1>1$, we have $\pi_1(V_{2n-2,n-1})=0$$\pi_1(V_{2n-2,n-1})=0$. Hence the standard framing of $S^1\times-1_{n-1}$$S^1\times-1_{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 a(D^2)$$\overline a(D^2)$ in $\Rr^{2n}$$\Rr^{2n}$. Thus $\overline a$$\overline a$ extends to an embedding

$\displaystyle \widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}.$ $$\displaystyle \widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}.$$

Define an embedding $g:S^n\to C_{f_0}$$g:S^n\to C_{f_0}$ by setting

$\displaystyle g(S^n):\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n$ $$\displaystyle g(S^n):\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n$$

with natural orientation.

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

Proposition 4.1. Linked embedded connected sum, or parametric connected sum, 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)$, unless $n=3$$n=3$ in the smooth category.

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

5 The Whitney invariant

Let $N$$N$ be a closed $n$$n$-manifold and fix an embedding $f_0:N\to\Rr^m$$f_0:N\to\Rr^m$. For any other embedding $f \colon N \to \Rr^m$$f \colon N \to \Rr^m$, there is an invariant $W(f, f_0)$$W(f, f_0)$, called the Whitney invariant, which we define in this section. Roughly speaking, $W(f,f_0)=W_{f_0}(f)=W(f)$$W(f,f_0)=W_{f_0}(f)=W(f)$ is defined as the homology class of the self-intersection set $\Sigma(H)$$\Sigma(H)$ of a general position homotopy $H$$H$ between $f$$f$ and $f_0$$f_0$. This is formalized in Definition 5.2 in the smooth category, following [Skopenkov2010]. The definition in the PL category is analogous [Hudson1969, $\S$$\S$12], [Vrabec1977, p. 145], [Skopenkov2006, $\S$$\S$2.4]. We begin by presenting a simpler definition of the Whitney invariant, Definition 5.1, for a particular case. For Theorem 2.1 only the case $m=2n$$m=2n$ is required.

Fix an orientation on $\Rr^m$$\Rr^m$. Assume that either $m-n$$m-n$ is even or $N$$N$ is oriented.

Definition 5.1. Assume that $N$$N$ is $(2n-m)$$(2n-m)$-connected and $2m\ge3n+3$$2m\ge3n+3$. Then restrictions of $f$$f$ and $f_0$$f_0$ to $N_0$$N_0$ are regular homotopic [Hirsch1959]. 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, cf. [Haefliger&Hirsch1963, 3.1.b], [Takase2006, Lemma 2.2]. 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$$n+2(n+1)<2m$, by general position $Cl(f\cap F)$$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. For $m-n$$m-n$ odd it has a natural orientation defined below, and so carries a homology class with $\Zz$$\Zz$ coefficients. Define $W(f)$$W(f)$ to be the homology class:

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

The orientation on $f\cap F$$f\cap F$ (extendable to $Cl(f\cap F)$$Cl(f\cap F)$) is defined for $m-n$$m-n$ odd as follows. For each point $x\in f\cap F$$x\in f\cap F$ take a vector at $x$$x$ tangent to $f\cap F$$f\cap F$. Complete this vector to a positive base tangent to $N$$N$. Since $n+2(n+1)<2m$$n+2(n+1)<2m$, by general position there is a unique point $y\in B^n\times I$$y\in B^n\times I$ such that $Fy=fx$$Fy=fx$. The tangent vector at $x$$x$ thus gives a tangent vector at $y$$y$ to $B^n\times I$$B^n\times I$. Complete this vector to a positive base tangent to $B^n\times I$$B^n\times I$, where the orientation on $B^n$$B^n$ comes from $N$$N$. The union of the images of the constructed two bases is a base at $Fy=fx$$Fy=fx$ of $\Rr^m$$\Rr^m$. If the latter base is positive, then call the initial vector 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 the latter base of $\Rr^m$$\Rr^m$, this condition indeed defines an orientation on $f\cap F$$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$. Since $m\ge n+2$$m\ge n+2$, by general position the closure $Cl\Sigma(H)$$Cl\Sigma(H)$ of the self-intersection set has codimension 2 singularities. So the closure carries a homology class with $\Zz_2$$\Zz_2$ coefficients. (Note that $Cl\Sigma(H)$$Cl\Sigma(H)$ can be assumed to be a submanifold for $2m\ge3n+2$$2m\ge3n+2$.) For $m-n$$m-n$ odd it has a natural orientation ($\S$$\S$7) and so carries a homology class with $\Zz$$\Zz$ coefficients. Define the Whitney invariant to be the homology class:

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

(The orientation is defined for each $m,n$$m,n$ but used only for odd $m-n$$m-n$. When $m-n$$m-n$ is even, for $W(f)$$W(f)$ being well-defined we need $\Zz_2$$\Zz_2$-coefficients.)

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

$\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_{\varepsilon(m-n-1)}),\quad [f] \mapsto W(f) = W(f, f_0).$ $$\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_{\varepsilon(m-n-1)}),\quad [f] \mapsto W(f) = W(f, f_0).$$

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 $H$$H$, which is proved analogously to [Skopenkov2006, $\S$$\S$2.4].

(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 $F$$F$ to be fixed on $N_0$$N_0$.) 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) Since a change of the orientation on $N$$N$ forces a change of the orientation on $B^n$$B^n$, the class $W(f)$$W(f)$ is independent of the choice of the orientation on $N$$N$. 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 we may assume that $f=f_0=\sigma\circ f$$f=f_0=\sigma\circ f$ on $N_0$$N_0$ and because a change of the orientation of $\Rr^m$$\Rr^m$ forces a change of the orientation of $f\cap F$$f\cap F$).

(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$, is $(a\mod2,0)$$(a\mod2,0)$ for $n=2$$n=2$.

(e) $W(f\#g)=W(f)$$W(f\#g)=W(f)$ for each pair of embeddings $f:N\to\Rr^m$$f:N\to\Rr^m$ and $g:S^n\to\Rr^m$$g:S^n\to\Rr^m$.

(f) For $m=2n+1$$m=2n+1$ the Whitney invariant can be recovered from the collection of pairwise linking coefficients of the components of $N$$N$, cf. Remark 3.2.b of [Skopenkov2016h].

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] 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 (Linked embedded connected sum, cf. [Skopenkov2010, Definition 1.4]). If $N$$N$ is $k$$k$-connected, then for an embedding $f_0:N\to S^{2n-k}$$f_0:N\to S^{2n-k}$ and a class $a\in H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$$a\in H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$ one can construct an embedding $f_a:N\to S^{2n-k}$$f_a:N\to S^{2n-k}$ by linked connected sum analogously to the case $k=0$$k=0$.

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$$n\ge2k+4$ in the PL or smooth categories, respectively). If $n=2k+3$$n=2k+3$, then this construction gives only a construction of embeddings $f_a:N\to\Rr^{2n-k}$$f_a:N\to\Rr^{2n-k}$ for each $a\in H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$$a\in H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$ but not a well-defined action of $H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$$H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$ on $E^{2n-k}(N)$$E^{2n-k}(N)$.

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

6.2 Classification just below the stable range

Theorem 6.2. Let $N$$N$ be a closed orientable homologically $k$$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\ge k+3$$n\ge k+3$ in the smooth category or $n\ge2k+4$$n\ge2k+4$ in the categoriy.

This was proved for $k$$k$-connected manifolds in the smooth category [Haefliger&Hirsch1963], and in the PL category in [Hudson1969, $\S$$\S$11], [Boechat&Haefliger1970], [Boechat1971], cf. [Vrabec1977]. The proof of [Haefliger&Hirsch1963], [Boechat&Haefliger1970], [Boechat1971] actually used the homological $k$$k$-connectedness assumption.

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].

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

By Theorem 6.2 the Whitney invariant $W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{\varepsilon(q)}$$W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{\varepsilon(q)}$ is bijective for $1\le p\le q-2$$1\le p\le q-2$. It is in fact a group isomorphism (for the group structure introduced in [Skopenkov2006a], [Skopenkov2015], [Skopenkov2015a]). The Hudson torus $\Hud(1)$$\Hud(1)$ generates $E^{p+2q+1}(S^p\times S^q)$$E^{p+2q+1}(S^p\times S^q)$ for $1\le p<q-1$$1\le p<q-1$; this holds by Theorem 6.2 because $W(\Hud(1),\Hud(0))=1\in\Zz_{\varepsilon(q)}$$W(\Hud(1),\Hud(0))=1\in\Zz_{\varepsilon(q)}$. Also, for $q\ge2$$q\ge2$ by Theorem 6.2 the Whitney invariants

$\displaystyle W^{3q}_{q-1,q}:E^{3q}_{PL}(S^{q-1}\times S^q)\to\Zz_{\varepsilon(q)} \quad\text{and}\quad W^{3q+1}_{q,q}:E^{3q+1}_{PL}(S^q\times S^q)\to\Zz_{\varepsilon(q)}\oplus \Zz_{\varepsilon(q)}$ $$\displaystyle W^{3q}_{q-1,q}:E^{3q}_{PL}(S^{q-1}\times S^q)\to\Zz_{\varepsilon(q)} \quad\text{and}\quad W^{3q+1}_{q,q}:E^{3q+1}_{PL}(S^q\times S^q)\to\Zz_{\varepsilon(q)}\oplus \Zz_{\varepsilon(q)}$$

are bijective. In the smooth category for $q$$q$ even $W^{3q}_{q-1,q}$$W^{3q}_{q-1,q}$ is not injective (see the next subsection), and $W^{3q+1}_{q,q}$$W^{3q+1}_{q,q}$ is not surjective [Boechat1971], [Skopenkov2016f], and $W^7_{2,2}$$W^7_{2,2}$ is not injective [Skopenkov2016f].

6.3 Classification in the presence of smoothly knotted spheres

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.

$\displaystyle E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)}$ $$\displaystyle E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)}$$

for each $s>1$$s>1$ [Haefliger1966]. The following result for $n=2k+3$$n=2k+3$ was obtained using the Boéchat-Haefliger formula for the smoothing obstruction [Boechat&Haefliger1970], [Boechat1971]. Using that formula one can define the higher-dimensional Kreck invariant [Skopenkov2008].

Theorem 6.3 [Skopenkov2008]. 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 each $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\}$.

E.g. by Theorem 6.3 the Whitney invariant $W:E^{6l}_D(S^{2l-1}\times S^{2l})\to\Zz$$W:E^{6l}_D(S^{2l-1}\times S^{2l})\to\Zz$ is surjective and for each $u\in\Zz$$u\in\Zz$ there is a 1-1 correspondence $W^{-1}u\to\Zz_u$$W^{-1}u\to\Zz_u$.

6.4 Classification further below the stable range

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 remarks on $E^{2n-1}(N)$$E^{2n-1}(N)$ in $\S$$\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]. 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

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

For $k=0$$k=0$ this is the same as General Position Theorem 2.1 [Skopenkov2016c] (because $V_{2n+1,n+1}$$V_{2n+1,n+1}$ is $(n-1)$$(n-1)$-connected). For $k=1$$k=1$ this is covered by Theorem 6.2; for $k\ge2$$k\ge2$ it is not.

E.g. by Theorem 6.4 there is a 1-1 correspondence $E^m(S^p\times S^q)\to\pi_p(V_{m,p+q+1})\oplus\pi_q(V_{m,p+q+1})$$E^m(S^p\times S^q)\to\pi_p(V_{m,p+q+1})\oplus\pi_q(V_{m,p+q+1})$ for $m\ge2q+3$$m\ge2q+3$ and $2m\ge3q+3p+4$$2m\ge3q+3p+4$. For a generalization see [Skopenkov2016k] (to knotted tori) and [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 each $M>n$$M>n$.

7 An orientation on the self-intersection set

Let $f:N\to\Rr^m$$f:N\to\Rr^m$ be a general position smooth map of an oriented $n$$n$-manifold $N$$N$. Assume that $m\ge n+2$$m\ge n+2$ so that the closure $Cl\Sigma(f)$$Cl\Sigma(f)$ of the self-intersection set of $f$$f$ has codimension 2 singularities.

Definition 7.1 (a natural orientation on $\Sigma(f)$$\Sigma(f)$). Take points $x,y\in N$$x,y\in N$ away from the singularities of $\Sigma(f)$$\Sigma(f)$ and such that $fx=fy$$fx=fy$. Then a $(2n-m)$$(2n-m)$-base $\xi_x$$\xi_x$ tangent to $\Sigma(f)$$\Sigma(f)$ at $x$$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)$$\Sigma(f)$ at $y$$y$. Since $N$$N$ is orientable, we can take positive $(m-n)$$(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$.

We remark that

• a change of the orientation of $N$$N$ forces changes of the signs of $\eta_x$$\eta_x$ and $\eta_y$$\eta_y$ and so does not change the orientation of $\Sigma(f)$$\Sigma(f)$.

• the natural orientation on $\Sigma(f)$$\Sigma(f)$ need not extend to $Cl\Sigma(f)$$Cl\Sigma(f)$: take the cone $D^3\to\Rr^5$$D^3\to\Rr^5$ over a general position map $S^2\to\Rr^4$$S^2\to\Rr^4$ having only one self-intersection point.

• the natural orientation on $\Sigma(f)$$\Sigma(f)$ extends to $Cl\Sigma(f)$$Cl\Sigma(f)$ if $m-n$$m-n$ is odd [Hudson1969, Lemma 11.4].

Definition 7.2 (a natural orientation on $f\Sigma(f)$$f\Sigma(f)$ for $m-n$$m-n$ even). Take a $(2n-m)$$(2n-m)$-base $\xi$$\xi$ at a point $x\in f\Sigma(f)$$x\in f\Sigma(f)$ away from the singularities of $f\Sigma(f)$$f\Sigma(f)$. Since $N$$N$ is orientable, we can take a positive $(m-n)$$(m-n)$-base $\eta_+$$\eta_+$ normal to $f\Sigma(f)$$f\Sigma(f)$ in one sheet of $f(N)$$f(N)$. Analogously construct an $(m-n)$$(m-n)$-base $\eta_-$$\eta_-$ for the other sheet of $f(N)$$f(N)$. Since $m-n$$m-n$ is even, the orientation of the base $(\xi,\eta_+,\eta_-)$$(\xi,\eta_+,\eta_-)$ of $\Rr^m$$\Rr^m$ does not depend on choosing the first and the other sheet of $f(N)$$f(N)$. 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)$$f\Sigma(f)$.

8 References

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\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$$N \to\Rr^m$ are isotopic [Skopenkov2016c, Theorem 3.2]. In this page we summarize the situation for $m=2n\ge6$$m=2n\ge6$ and some more general situations.

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].

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, either $n$$n$ is odd or $N$$N$ is orientable, and either $n\ge4$$n\ge4$ or $n=3$$n=3$ and we are in the PL category. 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.

This is proved in [Haefliger1962b, 1.3.e], [Haefliger1963], [Haefliger&Hirsch1963, Theorem 2.4], [Bausum1975, Theorem 43] in the smooth category, and in [Weber1967], [Vrabec1977, Theorem 1.1] in the PL category.

If $n\ge4$$n\ge4$ is even and $N$$N$ is a closed connected non-orientable $n$$n$-manifold, then [Bausum1975, Theorem 43] asserts that there is a 1-1 correspondence

$\displaystyle E^{2n}(N)\to \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s$ $$\displaystyle E^{2n}(N)\to \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s$$

(because $Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$$Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$ is given by multiplication with $w_1(N)$$w_1(N)$). This 1-1 correspondence can presumably be defined as a generalized Whitney invariant, but the proof used the Haefliger-Wu invariant whose definition can be found in [Skopenkov2006, $\S$$\S$5]. It would be interesting to check if this description of $E^{2n}(N)$$E^{2n}(N)$ is equivalent to its different forms [Haefliger1962b, 1.3.e], [Haefliger1963], [Vrabec1977, Theorem 1.1].

The classification of smooth embeddings of 3-manifolds in $\Rr^6$$\Rr^6$ is more complicated, see $\S$$\S$6.3 or [Skopenkov2016t].

For embeddings of $n$$n$-manifolds in $\Rr^{2n-1}$$\Rr^{2n-1}$ see the case of 4-manifolds [Skopenkov2016f], [Yasui1984] for $n\ge5$$n\ge5$ and [Saeki1999], [Skopenkov2010], [Tonkonog2010] for non-closed manifolds.

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

3 Hudson tori

Together with the Haefliger knotted sphere [Skopenkov2016t], the examples of Hudson tori presented below were the first examples of embeddings in codimension greater than 2 which are not isotopic to the standard embedding defined below. (Hudson's construction [Hudson1963] was not as explicit as those below.)

For $m\ge n+2$$m\ge n+2$ we define the standard embedding $S^p\times S^{n-p}\to\Rr^m$$S^p\times S^{n-p}\to\Rr^m$ as the composition of the standard embeddings $S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}=\Rr^{n+2}\to\Rr^m$$S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}=\Rr^{n+2}\to\Rr^m$. (Note that in [Skopenkov2015], [Skopenkov2015a] a different embedding is called a standard embedding. Since these two embeddings are isotopic, no confusion would appear.)

Let $1_n:=(1,0,\ldots,0)\in S^n$$1_n:=(1,0,\ldots,0)\in S^n$.

Let us construct, for each $a\in\Zz$$a\in\Zz$ and $n\ge2$$n\ge2$, an embedding

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

The reader might first consider the case $n=2$$n=2$.

Definition 3.1. Take the standard embeddings $2D^{n+1}\times S^{n-1}\subset\Rr^{2n}$$2D^{n+1}\times S^{n-1}\subset\Rr^{2n}$ (where $2$$2$ means homothety with coefficient 2) and $\partial D^2\subset\partial D^{n+1}$$\partial D^2\subset\partial D^{n+1}$. Take the embedded sphere and embedded torus,

$\displaystyle 2\partial D^{n+1}\times-1_{n-1}\subset 2D^{n+1}\times S^{n-1}\subset\Rr^{2n}\quad\text{and}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$ $$\displaystyle 2\partial D^{n+1}\times-1_{n-1}\subset 2D^{n+1}\times S^{n-1}\subset\Rr^{2n}\quad\text{and}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$

Join them by an arc whose interior misses the two embedded manifolds. The Hudson torus $\Hud_n(1)$$\Hud_n(1)$ is the embedded connected sum of the two embeddings along this arc, compatible with the orientation. (Unlike the unlinked embedded connected sum [Skopenkov2016c] this is a linked embedded connected sum, i.e. the connected sum of two embeddings whose images are not contained in disjoint cubes.)

We remark that the construction in Definition 3.1 works for $n=1$$n=1$. This is not the case for the next construction in Definition 3.2.

Definition 3.2. For $a\in\Zz$$a\in\Zz$ we repeat the construction of Definition 3.1 replacing $2\partial D^{n+1}\times-1_{n-1}$$2\partial D^{n+1}\times-1_{n-1}$ by $|a|$$|a|$ copies $(1+\frac1k)\partial D^{n+1}\times-1_{n-1}$$(1+\frac1k)\partial D^{n+1}\times-1_{n-1}$ ($k=1,\dots,|a|$$k=1,\dots,|a|$) of $S^n$$S^n$. The copies are outside $D^{n+1}\times S^{n-1}$$D^{n+1}\times S^{n-1}$ and are `parallel' to $\partial D^{n+1}\times-1_{n-1}$$\partial D^{n+1}\times-1_{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 by tubes joining each $k$$k$-th copy to the $(k+1)$$(k+1)$-th copy. We obtain an embedding $g:S^n\to\Rr^{2n}$$g:S^n\to\Rr^{2n}$. Let $\Hud_n(a)$$\Hud_n(a)$ be the linked embedded connected sum of $g$$g$ with the embedding $\partial D^2\times S^{n-1}\subset\Rr^{2n}$$\partial D^2\times S^{n-1}\subset\Rr^{2n}$ from Definition 3.1.

Clearly, $\Hud_n(0)$$\Hud_n(0)$ is isotopic to the standard embedding.

The original motivation for Hudson was that $\Hud_n(1)$$\Hud_n(1)$ is not isotopic to $\Hud_n(0)$$\Hud_n(0)$ for each $n$$n$ (this is a particular case of Proposition 3.3 below).

One guesses 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 exists and is `realized' by the homotopy class of the map

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

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

Proposition 3.3. 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 3.3 follows by calculation of the Whitney invariant (Remark 5.3.e below) and, for $n$$n$ even, by Theorem 2.1.

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)$ isotopic to $\Hud_2(2)$$\Hud_2(2)$? It would also be interesting to find an explicit construction of an isotopy between $\Hud_{2k}(a)$$\Hud_{2k}(a)$ and $\Hud_{2k}(a+2)$$\Hud_{2k}(a+2)$, cf. [Vrabec1977, $\S$$\S$5].

Definition 3.4. Let us give, for $a\in\Zz$$a\in\Zz$ and $n\ge2$$n\ge2$, another construction of embeddings

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

Define a map $S^0\times S^{n-1}\to D^n$$S^0\times S^{n-1}\to D^n$ to be the constant $0\in D^n$$0\in D^n$ on one component $1\times S^{n-1}$$1\times S^{n-1}$ and the `standard inclusion' $\{-1\}\times S^{n-1}\to\partial D^n\subset D^n$$\{-1\}\times S^{n-1}\to\partial D^n\subset D^n$ on the other component. This map gives an embedding

$\displaystyle S^0\times S^{n-1}\to D^n\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$ $$\displaystyle S^0\times S^{n-1}\to D^n\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$

(See [Skopenkov2006, Figure 2.2]. The image of this embedding is the union of the standard $S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$$S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$ and the graph of the identity map in $S^{n-1}\times S^{n-1}\subset D^n\times S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$$S^{n-1}\times S^{n-1}\subset D^n\times S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$.)

Take any $x\in S^{n-1}$$x\in S^{n-1}$. The disk $D^{n+1}\times x$$D^{n+1}\times x$ intersects the image of this embedding at two points lying in $D^n\times x$$D^n\times x$, i.e., at the image of an embedding $S^0\times x\to D^n\times x$$S^0\times x\to D^n\times x$. Extend the latter embedding to an embedding $S^1\times x\to D^{n+1}\times x$$S^1\times x\to D^{n+1}\times x$. See [Skopenkov2006, Figure 2.3].) Thus we obtain the Hudson torus

$\displaystyle \Hud_n'(1):S^1\times S^{n-1}\overset h\to D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$ $$\displaystyle \Hud_n'(1):S^1\times S^{n-1}\overset h\to D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$

Here $h(e^{i\theta},y):=(y\cos\theta,\sin\theta,y)$$h(e^{i\theta},y):=(y\cos\theta,\sin\theta,y)$, where $D^{n+1}$$D^{n+1}$ is identified with $D^n\times D^1$$D^n\times D^1$.

The embedding $\Hud_n'(a)$$\Hud_n'(a)$ is obtained in the same way starting from a map $\varphi:\{-1\}\times S^{n-1}\to\partial D^n$$\varphi:\{-1\}\times S^{n-1}\to\partial D^n$ of degree $a$$a$ instead of the `standard inclusion'.

Remark 3.5. (a) The analogue of Proposition 3.3 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$$n\ge4$ and are PL isotopic for $n\ge3$$n\ge3$ [Skopenkov2006a]. It would be interesting to know if they are isotopic for $n=2$$n=2$, or if they are smoothly isotopic for $n=3$$n=3$.

(c) For $n=2$$n=2$ these construction give what we call the left Hudson torus. The right Hudson torus is constructed analogously. It is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of $S^1\times S^1$$S^1\times S^1$. The right and the left Hudson tori are not isotopic by Remark 5.3.e 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$ 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})$. There are versions of these constructions corresponding to Definition 3.4. For $p=0$$p=0$ this corresponds to the Zeeman construction [Skopenkov2016h] and its composition with the second unframed Kirby move. It would be interesting to know if the links $\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$$\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$ are isotopic, cf. [Skopenkov2015a, Remark 2.9.b]. These constructions can be further generalized [Skopenkov2016k].

4 Action by linked embedded connected sum

In this section we generalize the construction of the Hudson torus $\Hud(a)$$\Hud(a)$. For $n\ge3$$n\ge3$, a closed connected orientable $n$$n$-manifold $N$$N$, an embedding $f_0:N\to\Rr^{2n}$$f_0:N\to\Rr^{2n}$ and $a\in H_1(N;\Zz_{\varepsilon(n-1)})$$a\in H_1(N;\Zz_{\varepsilon(n-1)})$, we construct an embedding $f_a:N\to\Rr^{2n}$$f_a:N\to\Rr^{2n}$. This embedding is obtained by linked embedded connected sum of $f_0$$f_0$ with an $n$$n$-sphere representing the homology Alexander dual of $a$$a$.

More precisely, represent $a$$a$ by an embedding $a:S^1\to N$$a:S^1\to N$. 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}$. Identify $\nu_{f_0}^{-1}a(S^1)$$\nu_{f_0}^{-1}a(S^1)$ with $S^1\times S^{n-1}$$S^1\times S^{n-1}$. In the next paragraph we recall definition of embedded surgery on $S^1\times*\subset S^1\times S^{n-1}$$S^1\times*\subset S^1\times S^{n-1}$, which yields an embedding $g:S^n\to C_{f_0}$$g:S^n\to C_{f_0}$. Then we define $f_a$$f_a$ to be the (linked) embedded connected sum of $f_0$$f_0$ and $g$$g$ (along a certain arc joining their images).

Take a vector field on $S^1\times-1_{n-1}$$S^1\times-1_{n-1}$ normal to $S^1\times S^{n-1}$$S^1\times S^{n-1}$. Extend $S^1\times-1_{n-1}$$S^1\times-1_{n-1}$ along this vector field to a map $\overline a:D^2\to\Rr^{2n}$$\overline a:D^2\to\Rr^{2n}$. Since $2n>4$$2n>4$ and $n+2<2n$$n+2<2n$, by general position we may assume that $\overline a$$\overline a$ is an embedding and $\overline a(Int D^2)$$\overline a(Int D^2)$ misses $f_0(N)\cup S^1\times S^{n-1}$$f_0(N)\cup S^1\times S^{n-1}$. Since $n-1>1$$n-1>1$, we have $\pi_1(V_{2n-2,n-1})=0$$\pi_1(V_{2n-2,n-1})=0$. Hence the standard framing of $S^1\times-1_{n-1}$$S^1\times-1_{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 a(D^2)$$\overline a(D^2)$ in $\Rr^{2n}$$\Rr^{2n}$. Thus $\overline a$$\overline a$ extends to an embedding

$\displaystyle \widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}.$ $$\displaystyle \widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}.$$

Define an embedding $g:S^n\to C_{f_0}$$g:S^n\to C_{f_0}$ by setting

$\displaystyle g(S^n):\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n$ $$\displaystyle g(S^n):\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n$$

with natural orientation.

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

Proposition 4.1. Linked embedded connected sum, or parametric connected sum, 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)$, unless $n=3$$n=3$ in the smooth category.

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

5 The Whitney invariant

Let $N$$N$ be a closed $n$$n$-manifold and fix an embedding $f_0:N\to\Rr^m$$f_0:N\to\Rr^m$. For any other embedding $f \colon N \to \Rr^m$$f \colon N \to \Rr^m$, there is an invariant $W(f, f_0)$$W(f, f_0)$, called the Whitney invariant, which we define in this section. Roughly speaking, $W(f,f_0)=W_{f_0}(f)=W(f)$$W(f,f_0)=W_{f_0}(f)=W(f)$ is defined as the homology class of the self-intersection set $\Sigma(H)$$\Sigma(H)$ of a general position homotopy $H$$H$ between $f$$f$ and $f_0$$f_0$. This is formalized in Definition 5.2 in the smooth category, following [Skopenkov2010]. The definition in the PL category is analogous [Hudson1969, $\S$$\S$12], [Vrabec1977, p. 145], [Skopenkov2006, $\S$$\S$2.4]. We begin by presenting a simpler definition of the Whitney invariant, Definition 5.1, for a particular case. For Theorem 2.1 only the case $m=2n$$m=2n$ is required.

Fix an orientation on $\Rr^m$$\Rr^m$. Assume that either $m-n$$m-n$ is even or $N$$N$ is oriented.

Definition 5.1. Assume that $N$$N$ is $(2n-m)$$(2n-m)$-connected and $2m\ge3n+3$$2m\ge3n+3$. Then restrictions of $f$$f$ and $f_0$$f_0$ to $N_0$$N_0$ are regular homotopic [Hirsch1959]. 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, cf. [Haefliger&Hirsch1963, 3.1.b], [Takase2006, Lemma 2.2]. 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$$n+2(n+1)<2m$, by general position $Cl(f\cap F)$$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. For $m-n$$m-n$ odd it has a natural orientation defined below, and so carries a homology class with $\Zz$$\Zz$ coefficients. Define $W(f)$$W(f)$ to be the homology class:

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

The orientation on $f\cap F$$f\cap F$ (extendable to $Cl(f\cap F)$$Cl(f\cap F)$) is defined for $m-n$$m-n$ odd as follows. For each point $x\in f\cap F$$x\in f\cap F$ take a vector at $x$$x$ tangent to $f\cap F$$f\cap F$. Complete this vector to a positive base tangent to $N$$N$. Since $n+2(n+1)<2m$$n+2(n+1)<2m$, by general position there is a unique point $y\in B^n\times I$$y\in B^n\times I$ such that $Fy=fx$$Fy=fx$. The tangent vector at $x$$x$ thus gives a tangent vector at $y$$y$ to $B^n\times I$$B^n\times I$. Complete this vector to a positive base tangent to $B^n\times I$$B^n\times I$, where the orientation on $B^n$$B^n$ comes from $N$$N$. The union of the images of the constructed two bases is a base at $Fy=fx$$Fy=fx$ of $\Rr^m$$\Rr^m$. If the latter base is positive, then call the initial vector 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 the latter base of $\Rr^m$$\Rr^m$, this condition indeed defines an orientation on $f\cap F$$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$. Since $m\ge n+2$$m\ge n+2$, by general position the closure $Cl\Sigma(H)$$Cl\Sigma(H)$ of the self-intersection set has codimension 2 singularities. So the closure carries a homology class with $\Zz_2$$\Zz_2$ coefficients. (Note that $Cl\Sigma(H)$$Cl\Sigma(H)$ can be assumed to be a submanifold for $2m\ge3n+2$$2m\ge3n+2$.) For $m-n$$m-n$ odd it has a natural orientation ($\S$$\S$7) and so carries a homology class with $\Zz$$\Zz$ coefficients. Define the Whitney invariant to be the homology class:

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

(The orientation is defined for each $m,n$$m,n$ but used only for odd $m-n$$m-n$. When $m-n$$m-n$ is even, for $W(f)$$W(f)$ being well-defined we need $\Zz_2$$\Zz_2$-coefficients.)

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

$\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_{\varepsilon(m-n-1)}),\quad [f] \mapsto W(f) = W(f, f_0).$ $$\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_{\varepsilon(m-n-1)}),\quad [f] \mapsto W(f) = W(f, f_0).$$

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 $H$$H$, which is proved analogously to [Skopenkov2006, $\S$$\S$2.4].

(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 $F$$F$ to be fixed on $N_0$$N_0$.) 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) Since a change of the orientation on $N$$N$ forces a change of the orientation on $B^n$$B^n$, the class $W(f)$$W(f)$ is independent of the choice of the orientation on $N$$N$. 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 we may assume that $f=f_0=\sigma\circ f$$f=f_0=\sigma\circ f$ on $N_0$$N_0$ and because a change of the orientation of $\Rr^m$$\Rr^m$ forces a change of the orientation of $f\cap F$$f\cap F$).

(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$, is $(a\mod2,0)$$(a\mod2,0)$ for $n=2$$n=2$.

(e) $W(f\#g)=W(f)$$W(f\#g)=W(f)$ for each pair of embeddings $f:N\to\Rr^m$$f:N\to\Rr^m$ and $g:S^n\to\Rr^m$$g:S^n\to\Rr^m$.

(f) For $m=2n+1$$m=2n+1$ the Whitney invariant can be recovered from the collection of pairwise linking coefficients of the components of $N$$N$, cf. Remark 3.2.b of [Skopenkov2016h].

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] 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 (Linked embedded connected sum, cf. [Skopenkov2010, Definition 1.4]). If $N$$N$ is $k$$k$-connected, then for an embedding $f_0:N\to S^{2n-k}$$f_0:N\to S^{2n-k}$ and a class $a\in H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$$a\in H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$ one can construct an embedding $f_a:N\to S^{2n-k}$$f_a:N\to S^{2n-k}$ by linked connected sum analogously to the case $k=0$$k=0$.

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$$n\ge2k+4$ in the PL or smooth categories, respectively). If $n=2k+3$$n=2k+3$, then this construction gives only a construction of embeddings $f_a:N\to\Rr^{2n-k}$$f_a:N\to\Rr^{2n-k}$ for each $a\in H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$$a\in H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$ but not a well-defined action of $H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$$H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$ on $E^{2n-k}(N)$$E^{2n-k}(N)$.

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

6.2 Classification just below the stable range

Theorem 6.2. Let $N$$N$ be a closed orientable homologically $k$$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\ge k+3$$n\ge k+3$ in the smooth category or $n\ge2k+4$$n\ge2k+4$ in the categoriy.

This was proved for $k$$k$-connected manifolds in the smooth category [Haefliger&Hirsch1963], and in the PL category in [Hudson1969, $\S$$\S$11], [Boechat&Haefliger1970], [Boechat1971], cf. [Vrabec1977]. The proof of [Haefliger&Hirsch1963], [Boechat&Haefliger1970], [Boechat1971] actually used the homological $k$$k$-connectedness assumption.

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].

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

By Theorem 6.2 the Whitney invariant $W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{\varepsilon(q)}$$W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{\varepsilon(q)}$ is bijective for $1\le p\le q-2$$1\le p\le q-2$. It is in fact a group isomorphism (for the group structure introduced in [Skopenkov2006a], [Skopenkov2015], [Skopenkov2015a]). The Hudson torus $\Hud(1)$$\Hud(1)$ generates $E^{p+2q+1}(S^p\times S^q)$$E^{p+2q+1}(S^p\times S^q)$ for $1\le p<q-1$$1\le p<q-1$; this holds by Theorem 6.2 because $W(\Hud(1),\Hud(0))=1\in\Zz_{\varepsilon(q)}$$W(\Hud(1),\Hud(0))=1\in\Zz_{\varepsilon(q)}$. Also, for $q\ge2$$q\ge2$ by Theorem 6.2 the Whitney invariants

$\displaystyle W^{3q}_{q-1,q}:E^{3q}_{PL}(S^{q-1}\times S^q)\to\Zz_{\varepsilon(q)} \quad\text{and}\quad W^{3q+1}_{q,q}:E^{3q+1}_{PL}(S^q\times S^q)\to\Zz_{\varepsilon(q)}\oplus \Zz_{\varepsilon(q)}$ $$\displaystyle W^{3q}_{q-1,q}:E^{3q}_{PL}(S^{q-1}\times S^q)\to\Zz_{\varepsilon(q)} \quad\text{and}\quad W^{3q+1}_{q,q}:E^{3q+1}_{PL}(S^q\times S^q)\to\Zz_{\varepsilon(q)}\oplus \Zz_{\varepsilon(q)}$$

are bijective. In the smooth category for $q$$q$ even $W^{3q}_{q-1,q}$$W^{3q}_{q-1,q}$ is not injective (see the next subsection), and $W^{3q+1}_{q,q}$$W^{3q+1}_{q,q}$ is not surjective [Boechat1971], [Skopenkov2016f], and $W^7_{2,2}$$W^7_{2,2}$ is not injective [Skopenkov2016f].

6.3 Classification in the presence of smoothly knotted spheres

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.

$\displaystyle E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)}$ $$\displaystyle E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)}$$

for each $s>1$$s>1$ [Haefliger1966]. The following result for $n=2k+3$$n=2k+3$ was obtained using the Boéchat-Haefliger formula for the smoothing obstruction [Boechat&Haefliger1970], [Boechat1971]. Using that formula one can define the higher-dimensional Kreck invariant [Skopenkov2008].

Theorem 6.3 [Skopenkov2008]. 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 each $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\}$.

E.g. by Theorem 6.3 the Whitney invariant $W:E^{6l}_D(S^{2l-1}\times S^{2l})\to\Zz$$W:E^{6l}_D(S^{2l-1}\times S^{2l})\to\Zz$ is surjective and for each $u\in\Zz$$u\in\Zz$ there is a 1-1 correspondence $W^{-1}u\to\Zz_u$$W^{-1}u\to\Zz_u$.

6.4 Classification further below the stable range

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 remarks on $E^{2n-1}(N)$$E^{2n-1}(N)$ in $\S$$\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]. 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

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

For $k=0$$k=0$ this is the same as General Position Theorem 2.1 [Skopenkov2016c] (because $V_{2n+1,n+1}$$V_{2n+1,n+1}$ is $(n-1)$$(n-1)$-connected). For $k=1$$k=1$ this is covered by Theorem 6.2; for $k\ge2$$k\ge2$ it is not.

E.g. by Theorem 6.4 there is a 1-1 correspondence $E^m(S^p\times S^q)\to\pi_p(V_{m,p+q+1})\oplus\pi_q(V_{m,p+q+1})$$E^m(S^p\times S^q)\to\pi_p(V_{m,p+q+1})\oplus\pi_q(V_{m,p+q+1})$ for $m\ge2q+3$$m\ge2q+3$ and $2m\ge3q+3p+4$$2m\ge3q+3p+4$. For a generalization see [Skopenkov2016k] (to knotted tori) and [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 each $M>n$$M>n$.

7 An orientation on the self-intersection set

Let $f:N\to\Rr^m$$f:N\to\Rr^m$ be a general position smooth map of an oriented $n$$n$-manifold $N$$N$. Assume that $m\ge n+2$$m\ge n+2$ so that the closure $Cl\Sigma(f)$$Cl\Sigma(f)$ of the self-intersection set of $f$$f$ has codimension 2 singularities.

Definition 7.1 (a natural orientation on $\Sigma(f)$$\Sigma(f)$). Take points $x,y\in N$$x,y\in N$ away from the singularities of $\Sigma(f)$$\Sigma(f)$ and such that $fx=fy$$fx=fy$. Then a $(2n-m)$$(2n-m)$-base $\xi_x$$\xi_x$ tangent to $\Sigma(f)$$\Sigma(f)$ at $x$$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)$$\Sigma(f)$ at $y$$y$. Since $N$$N$ is orientable, we can take positive $(m-n)$$(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$.

We remark that

• a change of the orientation of $N$$N$ forces changes of the signs of $\eta_x$$\eta_x$ and $\eta_y$$\eta_y$ and so does not change the orientation of $\Sigma(f)$$\Sigma(f)$.

• the natural orientation on $\Sigma(f)$$\Sigma(f)$ need not extend to $Cl\Sigma(f)$$Cl\Sigma(f)$: take the cone $D^3\to\Rr^5$$D^3\to\Rr^5$ over a general position map $S^2\to\Rr^4$$S^2\to\Rr^4$ having only one self-intersection point.

• the natural orientation on $\Sigma(f)$$\Sigma(f)$ extends to $Cl\Sigma(f)$$Cl\Sigma(f)$ if $m-n$$m-n$ is odd [Hudson1969, Lemma 11.4].

Definition 7.2 (a natural orientation on $f\Sigma(f)$$f\Sigma(f)$ for $m-n$$m-n$ even). Take a $(2n-m)$$(2n-m)$-base $\xi$$\xi$ at a point $x\in f\Sigma(f)$$x\in f\Sigma(f)$ away from the singularities of $f\Sigma(f)$$f\Sigma(f)$. Since $N$$N$ is orientable, we can take a positive $(m-n)$$(m-n)$-base $\eta_+$$\eta_+$ normal to $f\Sigma(f)$$f\Sigma(f)$ in one sheet of $f(N)$$f(N)$. Analogously construct an $(m-n)$$(m-n)$-base $\eta_-$$\eta_-$ for the other sheet of $f(N)$$f(N)$. Since $m-n$$m-n$ is even, the orientation of the base $(\xi,\eta_+,\eta_-)$$(\xi,\eta_+,\eta_-)$ of $\Rr^m$$\Rr^m$ does not depend on choosing the first and the other sheet of $f(N)$$f(N)$. 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)$$f\Sigma(f)$.

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\times S^{n-1}$ and the `standard inclusion' $\{-1\}\times S^{n-1}\to\partial D^n\subset D^n$ on the other component. This map gives an ''embedding'' $$S^0\times S^{n-1}\to D^n\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$ (See \cite[Figure 2.2]{Skopenkov2006}. The image of this embedding is the union of the standard $S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$ and the graph of the identity map in $S^{n-1}\times S^{n-1}\subset D^n\times S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$.) Take any $x\in S^{n-1}$. The disk $D^{n+1}\times x$ intersects the image of this embedding at two points lying in $D^n\times x$, i.e., at the image of an embedding $S^0\times x\to D^n\times x$. Extend the latter embedding to an embedding $S^1\times x\to D^{n+1}\times x$. See \cite[Figure 2.3]{Skopenkov2006}.) Thus we obtain ''the Hudson torus'' $$\Hud_n'(1):S^1\times S^{n-1}\overset h\to D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$ Here $h(e^{i\theta},y):=(y\cos\theta,\sin\theta,y)$, where $D^{n+1}$ is identified with $D^n\times D^1$. The embedding $\Hud_n'(a)$ is obtained in the same way starting from a map $\varphi:\{-1\}\times S^{n-1}\to\partial D^n$ of degree $a$ instead of the `standard inclusion'. {{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 PL isotopic for $n\ge3$ \cite{Skopenkov2006a}. It would be interesting to know if they are isotopic for $n=2$, or if they are smoothly isotopic for $n=3$. (c) For $n=2$ these construction give 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}.e below. (d) Analogously one constructs the Hudson torus $\Hud_{n,p}(a):S^p\times S^{n-p}\to\Rr^{2n-p+1}$ for $a\in\Zz$ or, more generally, $\Hud_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$ for $a\in\pi_n(S^{m-n+p-1})$. There are versions of these constructions corresponding to Definition \ref{dh2n}. For $p=0$ this corresponds to [[High_codimension_links#Examples|the Zeeman construction]] \cite{Skopenkov2016h} and its composition with ''the second unframed Kirby move''. 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 2.9.b]{Skopenkov2015a}. These constructions can be [[Knotted_tori#Examples|further generalized]] \cite{Skopenkov2016k}. {{endthm}} ==Action by linked embedded connected sum== ; In this section we generalize the construction of the Hudson torus $\Hud(a)$. For $n\ge3$, a closed connected orientable $n$-manifold $N$, an embedding $f_0:N\to\Rr^{2n}$ and $a\in H_1(N;\Zz_{\varepsilon(n-1)})$, we construct an embedding $f_a:N\to\Rr^{2n}$. This embedding is obtained by linked embedded connected sum of $f_0$ with an $n$-sphere representing the homology Alexander dual of $a$. More precisely, represent $a$ by an embedding $a:S^1\to N$. Since any orientable bundle over $S^1$ is trivial, $\nu_{f_0}^{-1}a(S^1)\cong S^1\times S^{n-1}$. Identify $\nu_{f_0}^{-1}a(S^1)$ with $S^1\times S^{n-1}$. In the next paragraph we recall definition of embedded surgery on $S^1\times*\subset S^1\times S^{n-1}$, which yields an embedding $g:S^n\to C_{f_0}$. Then we define $f_a$ to be the (linked) embedded connected sum of $f_0$ and $g$ (along a certain arc joining their images). Take a vector field on $S^1\times-1_{n-1}$ normal to $S^1\times S^{n-1}$. Extend $S^1\times-1_{n-1}$ along this vector field to a map $\overline a:D^2\to\Rr^{2n}$. Since n>4$ and $n+2<2n$, by general position we may assume that $\overline a$ is an embedding and $\overline a(Int D^2)$ misses $f_0(N)\cup S^1\times S^{n-1}$. Since $n-1>1$, we have $\pi_1(V_{2n-2,n-1})=0$. Hence the standard framing of $S^1\times-1_{n-1}$ in $S^1\times S^{n-1}$ extends to an $(n-1)$-framing on $\overline a(D^2)$ in $\Rr^{2n}$. Thus $\overline a$ extends to an embedding $$ \widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}. $$ Define an embedding $g:S^n\to C_{f_0}$ by setting $$ g(S^n):\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n $$ with natural orientation. By Definition \ref{dw1} of [[#The_Whitney_invariant|the Whitney invariant]], $W(f_a,f_0)=a$. Thus by Theorem \ref{th4} 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} Linked embedded connected sum, or [[Parametric_connected_sum#Applications|parametric connected sum]], define free transitive actions of $H_1(N;\Zz_{\varepsilon(n-1)})$ on $E^{2n}(N)$, unless $n=3$ in the smooth category. {{endthm}} This follows by Theorem \ref{th4} or \cite[Remark 18.a]{Skopenkov2014}. == The Whitney invariant == ; \label{s:whitney} Let $N$ be a closed $n$-manifold and fix an embedding $f_0:N\to\Rr^m$. For any other embedding $f \colon N \to \Rr^m$, there is an invariant $W(f, f_0)$, called the Whitney invariant, which we define in this section. Roughly speaking, $W(f,f_0)=W_{f_0}(f)=W(f)$ is defined as the homology class of the self-intersection set $\Sigma(H)$ 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}. The definition in the PL category is analogous \cite[$\S]{Hudson1969}, \cite[p. 145]{Vrabec1977}, \cite[$\S.4]{Skopenkov2006}. We begin by presenting a simpler definition of the Whitney invariant, Definition \ref{dw1}, for a particular case. For Theorem \ref{th4} only the case $m=2n$ is required. Fix an orientation on $\Rr^m$. Assume that either $m-n$ is even or $N$ is oriented. {{beginthm|Definition}}\label{dw1} Assume that $N$ is $(2n-m)$-connected and m\ge3n+3$. Then restrictions of $f$ and $f_0$ to $N_0$ are regular homotopic \cite{Hirsch1959}. Since $N$ is $(2n-m)$-connected, $N_0$ retracts to an $(m-n-1)$-dimensional polyhedron. Therefore these restrictions are isotopic, cf. \cite[3.1.b]{Haefliger&Hirsch1963}, \cite[Lemma 2.2]{Takase2006}. 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 $Cl(f\cap F)$ 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. For $m-n$ odd it has a natural orientation defined below, and so carries a homology class with $\Zz$ coefficients. Define $W(f)$ to be the homology class: $$ W(f):=[Cl(f\cap F)]\in H_{2n-m+1}(N_0,\partial N_0;\Zz_{\varepsilon(m-n-1)})\cong H_{2n-m+1}(N;\Zz_{\varepsilon(m-n-1)}). $$ The orientation on $f\cap F$ (extendable to $Cl(f\cap F)$) is defined for $m-n$ odd as follows. For each point $x\in f\cap F$ take a vector at $x$ tangent to $f\cap F$. Complete this vector to a positive base tangent to $N$. Since $n+2(n+1)<2m$, by general position there is a unique point $y\in B^n\times I$ such that $Fy=fx$. The tangent vector at $x$ thus gives a tangent vector at $y$ to $B^n\times I$. Complete this vector to a positive base tangent to $B^n\times I$, where the orientation on $B^n$ comes from $N$. The union of the images of the constructed two bases is a base at $Fy=fx$ of $\Rr^m$. If the latter base is positive, then call the initial vector of $f\cap F$ ''positive''. Since a change of the orientation on $f\cap F$ forces a change of the orientation of the latter base of $\Rr^m$, 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 $f$. Since $m\ge n+2$, by general position the closure $Cl\Sigma(H)$ of the self-intersection set has codimension 2 singularities. So the closure carries a homology class with $\Zz_2$ coefficients. (Note that $Cl\Sigma(H)$ can be assumed to be a submanifold for m\ge3n+2$.) For $m-n$ odd it has a [[#An orientation on the self-intersection set|natural orientation]] ($\S) and so carries a homology class with $\Zz$ coefficients. Define the Whitney invariant to be the homology class: $$ W(f):=[Cl\Sigma(H)]\in H_{2n-m+1}(N\times I;\Zz_{\varepsilon(m-n-1)})\cong H_{2n-m+1}(N;\Zz_{\varepsilon(m-n-1)}). $$ (The orientation is defined for each $m,n$ but used only for odd $m-n$. When $m-n$ is even, for $W(f)$ being well-defined we need $\Zz_2$-coefficients.) Clearly, $W(f) = W(f')$ if $f$ is isotopic to $f'$. Hence the Whitney invariant defines a map $$W:E^m(N)\to H_{2n-m+1}(N;\Zz_{\varepsilon(m-n-1)}),\quad [f] \mapsto W(f) = W(f, f_0).$$ {{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 $H$, which is proved analogously to \cite[$\S.4]{Skopenkov2006}. (b) Definition \ref{dw1} is a particular case of Definition \ref{dw2}. (Indeed, if $f=f_0$ on $N_0$, we can take $F$ to be fixed on $N_0$.) 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) Since a change of the orientation on $N$ forces a change of the orientation on $B^n$, the class $W(f)$ is independent of the choice of the orientation on $N$. For the reflection $\sigma:\Rr^m\to\Rr^m$ with respect to a hyperplane we have $W(\sigma\circ f)=-W(f)$ (because we may assume that $f=f_0=\sigma\circ f$ on $N_0$ and because a change of the orientation of $\Rr^m$ forces a change of the orientation of $f\cap F$). (d) For the [[#Hudson tori|Hudson tori]] $W(\Hud_n(a))=W(\Hud'_n(a))$ is $a$ or $a\mod2$ for $n\ge3$, is $(a\mod2,0)$ for $n=2$. (e) $W(f\#g)=W(f)$ for each pair of embeddings $f:N\to\Rr^m$ and $g:S^n\to\Rr^m$. (f) For $m=2n+1$ the Whitney invariant can be recovered from the collection of pairwise linking coefficients of the components of $N$, cf. [[High_codimension_links#The linking coefficient|Remark 3.2.b]] of \cite{Skopenkov2016h}. {{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{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|(Linked embedded connected sum, cf. \cite[Definition 1.4]{Skopenkov2010})}}\label{hicoe} If $N$ is $k$-connected, then for an embedding $f_0:N\to S^{2n-k}$ and a class $a\in H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$ one can construct an embedding $f_a:N\to S^{2n-k}$ by linked connected sum analogously to [[#Action_by_linked_embedded_connected_sum|the case $k=0$]]. {{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). If $n=2k+3$, then this construction gives only a construction of embeddings $f_a:N\to\Rr^{2n-k}$ for each $a\in H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$ but not a well-defined action of $H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$ on $E^{2n-k}(N)$. The embedding $f_a$ has an alternative construction using [[Parametric_connected_sum#Applications|parametric connected sum]] \cite[Remark 18.a]{Skopenkov2014}. === Classification just below the stable range === ; \label{s:just} {{beginthm|Theorem}}\label{hico} Let $N$ be a closed orientable homologically $k$-connected $n$-manifold, $k\ge0$. Then [[Embeddings just below the stable range: classification#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 $n\ge k+3$ in the smooth category or $n\ge2k+4$ in the categoriy. {{endthm}} This was proved for $k$-connected manifolds in the smooth category \cite{Haefliger&Hirsch1963}, and in the PL category in \cite[$\S]{Hudson1969}, \cite{Boechat&Haefliger1970}, \cite{Boechat1971}, cf. \cite{Vrabec1977}. The proof of \cite{Haefliger&Hirsch1963}, \cite{Boechat&Haefliger1970}, \cite{Boechat1971} actually used the homological $k$-connectedness assumption. 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{Skopenkov2016c}. An inverse to the map $W$ of Theorem \ref{hico} is given by Example \ref{hicoe}. By Theorem \ref{hico} the Whitney invariant $W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{\varepsilon(q)}$ is bijective for 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$$N \to\Rr^m$ are isotopic [Skopenkov2016c, Theorem 3.2]. In this page we summarize the situation for $m=2n\ge6$$m=2n\ge6$ and some more general situations.

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].

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, either $n$$n$ is odd or $N$$N$ is orientable, and either $n\ge4$$n\ge4$ or $n=3$$n=3$ and we are in the PL category. 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.

This is proved in [Haefliger1962b, 1.3.e], [Haefliger1963], [Haefliger&Hirsch1963, Theorem 2.4], [Bausum1975, Theorem 43] in the smooth category, and in [Weber1967], [Vrabec1977, Theorem 1.1] in the PL category.

If $n\ge4$$n\ge4$ is even and $N$$N$ is a closed connected non-orientable $n$$n$-manifold, then [Bausum1975, Theorem 43] asserts that there is a 1-1 correspondence

$\displaystyle E^{2n}(N)\to \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s$ $$\displaystyle E^{2n}(N)\to \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s$$

(because $Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$$Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$ is given by multiplication with $w_1(N)$$w_1(N)$). This 1-1 correspondence can presumably be defined as a generalized Whitney invariant, but the proof used the Haefliger-Wu invariant whose definition can be found in [Skopenkov2006, $\S$$\S$5]. It would be interesting to check if this description of $E^{2n}(N)$$E^{2n}(N)$ is equivalent to its different forms [Haefliger1962b, 1.3.e], [Haefliger1963], [Vrabec1977, Theorem 1.1].

The classification of smooth embeddings of 3-manifolds in $\Rr^6$$\Rr^6$ is more complicated, see $\S$$\S$6.3 or [Skopenkov2016t].

For embeddings of $n$$n$-manifolds in $\Rr^{2n-1}$$\Rr^{2n-1}$ see the case of 4-manifolds [Skopenkov2016f], [Yasui1984] for $n\ge5$$n\ge5$ and [Saeki1999], [Skopenkov2010], [Tonkonog2010] for non-closed manifolds.

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

3 Hudson tori

Together with the Haefliger knotted sphere [Skopenkov2016t], the examples of Hudson tori presented below were the first examples of embeddings in codimension greater than 2 which are not isotopic to the standard embedding defined below. (Hudson's construction [Hudson1963] was not as explicit as those below.)

For $m\ge n+2$$m\ge n+2$ we define the standard embedding $S^p\times S^{n-p}\to\Rr^m$$S^p\times S^{n-p}\to\Rr^m$ as the composition of the standard embeddings $S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}=\Rr^{n+2}\to\Rr^m$$S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}=\Rr^{n+2}\to\Rr^m$. (Note that in [Skopenkov2015], [Skopenkov2015a] a different embedding is called a standard embedding. Since these two embeddings are isotopic, no confusion would appear.)

Let $1_n:=(1,0,\ldots,0)\in S^n$$1_n:=(1,0,\ldots,0)\in S^n$.

Let us construct, for each $a\in\Zz$$a\in\Zz$ and $n\ge2$$n\ge2$, an embedding

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

The reader might first consider the case $n=2$$n=2$.

Definition 3.1. Take the standard embeddings $2D^{n+1}\times S^{n-1}\subset\Rr^{2n}$$2D^{n+1}\times S^{n-1}\subset\Rr^{2n}$ (where $2$$2$ means homothety with coefficient 2) and $\partial D^2\subset\partial D^{n+1}$$\partial D^2\subset\partial D^{n+1}$. Take the embedded sphere and embedded torus,

$\displaystyle 2\partial D^{n+1}\times-1_{n-1}\subset 2D^{n+1}\times S^{n-1}\subset\Rr^{2n}\quad\text{and}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$ $$\displaystyle 2\partial D^{n+1}\times-1_{n-1}\subset 2D^{n+1}\times S^{n-1}\subset\Rr^{2n}\quad\text{and}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$

Join them by an arc whose interior misses the two embedded manifolds. The Hudson torus $\Hud_n(1)$$\Hud_n(1)$ is the embedded connected sum of the two embeddings along this arc, compatible with the orientation. (Unlike the unlinked embedded connected sum [Skopenkov2016c] this is a linked embedded connected sum, i.e. the connected sum of two embeddings whose images are not contained in disjoint cubes.)

We remark that the construction in Definition 3.1 works for $n=1$$n=1$. This is not the case for the next construction in Definition 3.2.

Definition 3.2. For $a\in\Zz$$a\in\Zz$ we repeat the construction of Definition 3.1 replacing $2\partial D^{n+1}\times-1_{n-1}$$2\partial D^{n+1}\times-1_{n-1}$ by $|a|$$|a|$ copies $(1+\frac1k)\partial D^{n+1}\times-1_{n-1}$$(1+\frac1k)\partial D^{n+1}\times-1_{n-1}$ ($k=1,\dots,|a|$$k=1,\dots,|a|$) of $S^n$$S^n$. The copies are outside $D^{n+1}\times S^{n-1}$$D^{n+1}\times S^{n-1}$ and are `parallel' to $\partial D^{n+1}\times-1_{n-1}$$\partial D^{n+1}\times-1_{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 by tubes joining each $k$$k$-th copy to the $(k+1)$$(k+1)$-th copy. We obtain an embedding $g:S^n\to\Rr^{2n}$$g:S^n\to\Rr^{2n}$. Let $\Hud_n(a)$$\Hud_n(a)$ be the linked embedded connected sum of $g$$g$ with the embedding $\partial D^2\times S^{n-1}\subset\Rr^{2n}$$\partial D^2\times S^{n-1}\subset\Rr^{2n}$ from Definition 3.1.

Clearly, $\Hud_n(0)$$\Hud_n(0)$ is isotopic to the standard embedding.

The original motivation for Hudson was that $\Hud_n(1)$$\Hud_n(1)$ is not isotopic to $\Hud_n(0)$$\Hud_n(0)$ for each $n$$n$ (this is a particular case of Proposition 3.3 below).

One guesses 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 exists and is `realized' by the homotopy class of the map

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

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

Proposition 3.3. 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 3.3 follows by calculation of the Whitney invariant (Remark 5.3.e below) and, for $n$$n$ even, by Theorem 2.1.

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)$ isotopic to $\Hud_2(2)$$\Hud_2(2)$? It would also be interesting to find an explicit construction of an isotopy between $\Hud_{2k}(a)$$\Hud_{2k}(a)$ and $\Hud_{2k}(a+2)$$\Hud_{2k}(a+2)$, cf. [Vrabec1977, $\S$$\S$5].

Definition 3.4. Let us give, for $a\in\Zz$$a\in\Zz$ and $n\ge2$$n\ge2$, another construction of embeddings

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

Define a map $S^0\times S^{n-1}\to D^n$$S^0\times S^{n-1}\to D^n$ to be the constant $0\in D^n$$0\in D^n$ on one component $1\times S^{n-1}$$1\times S^{n-1}$ and the `standard inclusion' $\{-1\}\times S^{n-1}\to\partial D^n\subset D^n$$\{-1\}\times S^{n-1}\to\partial D^n\subset D^n$ on the other component. This map gives an embedding

$\displaystyle S^0\times S^{n-1}\to D^n\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$ $$\displaystyle S^0\times S^{n-1}\to D^n\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$

(See [Skopenkov2006, Figure 2.2]. The image of this embedding is the union of the standard $S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$$S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$ and the graph of the identity map in $S^{n-1}\times S^{n-1}\subset D^n\times S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$$S^{n-1}\times S^{n-1}\subset D^n\times S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$.)

Take any $x\in S^{n-1}$$x\in S^{n-1}$. The disk $D^{n+1}\times x$$D^{n+1}\times x$ intersects the image of this embedding at two points lying in $D^n\times x$$D^n\times x$, i.e., at the image of an embedding $S^0\times x\to D^n\times x$$S^0\times x\to D^n\times x$. Extend the latter embedding to an embedding $S^1\times x\to D^{n+1}\times x$$S^1\times x\to D^{n+1}\times x$. See [Skopenkov2006, Figure 2.3].) Thus we obtain the Hudson torus

$\displaystyle \Hud_n'(1):S^1\times S^{n-1}\overset h\to D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$ $$\displaystyle \Hud_n'(1):S^1\times S^{n-1}\overset h\to D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$

Here $h(e^{i\theta},y):=(y\cos\theta,\sin\theta,y)$$h(e^{i\theta},y):=(y\cos\theta,\sin\theta,y)$, where $D^{n+1}$$D^{n+1}$ is identified with $D^n\times D^1$$D^n\times D^1$.

The embedding $\Hud_n'(a)$$\Hud_n'(a)$ is obtained in the same way starting from a map $\varphi:\{-1\}\times S^{n-1}\to\partial D^n$$\varphi:\{-1\}\times S^{n-1}\to\partial D^n$ of degree $a$$a$ instead of the `standard inclusion'.

Remark 3.5. (a) The analogue of Proposition 3.3 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$$n\ge4$ and are PL isotopic for $n\ge3$$n\ge3$ [Skopenkov2006a]. It would be interesting to know if they are isotopic for $n=2$$n=2$, or if they are smoothly isotopic for $n=3$$n=3$.

(c) For $n=2$$n=2$ these construction give what we call the left Hudson torus. The right Hudson torus is constructed analogously. It is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of $S^1\times S^1$$S^1\times S^1$. The right and the left Hudson tori are not isotopic by Remark 5.3.e 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$ 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})$. There are versions of these constructions corresponding to Definition 3.4. For $p=0$$p=0$ this corresponds to the Zeeman construction [Skopenkov2016h] and its composition with the second unframed Kirby move. It would be interesting to know if the links $\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$$\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$ are isotopic, cf. [Skopenkov2015a, Remark 2.9.b]. These constructions can be further generalized [Skopenkov2016k].

4 Action by linked embedded connected sum

In this section we generalize the construction of the Hudson torus $\Hud(a)$$\Hud(a)$. For $n\ge3$$n\ge3$, a closed connected orientable $n$$n$-manifold $N$$N$, an embedding $f_0:N\to\Rr^{2n}$$f_0:N\to\Rr^{2n}$ and $a\in H_1(N;\Zz_{\varepsilon(n-1)})$$a\in H_1(N;\Zz_{\varepsilon(n-1)})$, we construct an embedding $f_a:N\to\Rr^{2n}$$f_a:N\to\Rr^{2n}$. This embedding is obtained by linked embedded connected sum of $f_0$$f_0$ with an $n$$n$-sphere representing the homology Alexander dual of $a$$a$.

More precisely, represent $a$$a$ by an embedding $a:S^1\to N$$a:S^1\to N$. 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}$. Identify $\nu_{f_0}^{-1}a(S^1)$$\nu_{f_0}^{-1}a(S^1)$ with $S^1\times S^{n-1}$$S^1\times S^{n-1}$. In the next paragraph we recall definition of embedded surgery on $S^1\times*\subset S^1\times S^{n-1}$$S^1\times*\subset S^1\times S^{n-1}$, which yields an embedding $g:S^n\to C_{f_0}$$g:S^n\to C_{f_0}$. Then we define $f_a$$f_a$ to be the (linked) embedded connected sum of $f_0$$f_0$ and $g$$g$ (along a certain arc joining their images).

Take a vector field on $S^1\times-1_{n-1}$$S^1\times-1_{n-1}$ normal to $S^1\times S^{n-1}$$S^1\times S^{n-1}$. Extend $S^1\times-1_{n-1}$$S^1\times-1_{n-1}$ along this vector field to a map $\overline a:D^2\to\Rr^{2n}$$\overline a:D^2\to\Rr^{2n}$. Since $2n>4$$2n>4$ and $n+2<2n$$n+2<2n$, by general position we may assume that $\overline a$$\overline a$ is an embedding and $\overline a(Int D^2)$$\overline a(Int D^2)$ misses $f_0(N)\cup S^1\times S^{n-1}$$f_0(N)\cup S^1\times S^{n-1}$. Since $n-1>1$$n-1>1$, we have $\pi_1(V_{2n-2,n-1})=0$$\pi_1(V_{2n-2,n-1})=0$. Hence the standard framing of $S^1\times-1_{n-1}$$S^1\times-1_{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 a(D^2)$$\overline a(D^2)$ in $\Rr^{2n}$$\Rr^{2n}$. Thus $\overline a$$\overline a$ extends to an embedding

$\displaystyle \widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}.$ $$\displaystyle \widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}.$$

Define an embedding $g:S^n\to C_{f_0}$$g:S^n\to C_{f_0}$ by setting

$\displaystyle g(S^n):\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n$ $$\displaystyle g(S^n):\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n$$

with natural orientation.

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

Proposition 4.1. Linked embedded connected sum, or parametric connected sum, 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)$, unless $n=3$$n=3$ in the smooth category.

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

5 The Whitney invariant

Let $N$$N$ be a closed $n$$n$-manifold and fix an embedding $f_0:N\to\Rr^m$$f_0:N\to\Rr^m$. For any other embedding $f \colon N \to \Rr^m$$f \colon N \to \Rr^m$, there is an invariant $W(f, f_0)$$W(f, f_0)$, called the Whitney invariant, which we define in this section. Roughly speaking, $W(f,f_0)=W_{f_0}(f)=W(f)$$W(f,f_0)=W_{f_0}(f)=W(f)$ is defined as the homology class of the self-intersection set $\Sigma(H)$$\Sigma(H)$ of a general position homotopy $H$$H$ between $f$$f$ and $f_0$$f_0$. This is formalized in Definition 5.2 in the smooth category, following [Skopenkov2010]. The definition in the PL category is analogous [Hudson1969, $\S$$\S$12], [Vrabec1977, p. 145], [Skopenkov2006, $\S$$\S$2.4]. We begin by presenting a simpler definition of the Whitney invariant, Definition 5.1, for a particular case. For Theorem 2.1 only the case $m=2n$$m=2n$ is required.

Fix an orientation on $\Rr^m$$\Rr^m$. Assume that either $m-n$$m-n$ is even or $N$$N$ is oriented.

Definition 5.1. Assume that $N$$N$ is $(2n-m)$$(2n-m)$-connected and $2m\ge3n+3$$2m\ge3n+3$. Then restrictions of $f$$f$ and $f_0$$f_0$ to $N_0$$N_0$ are regular homotopic [Hirsch1959]. 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, cf. [Haefliger&Hirsch1963, 3.1.b], [Takase2006, Lemma 2.2]. 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$$n+2(n+1)<2m$, by general position $Cl(f\cap F)$$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. For $m-n$$m-n$ odd it has a natural orientation defined below, and so carries a homology class with $\Zz$$\Zz$ coefficients. Define $W(f)$$W(f)$ to be the homology class:

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

The orientation on $f\cap F$$f\cap F$ (extendable to $Cl(f\cap F)$$Cl(f\cap F)$) is defined for $m-n$$m-n$ odd as follows. For each point $x\in f\cap F$$x\in f\cap F$ take a vector at $x$$x$ tangent to $f\cap F$$f\cap F$. Complete this vector to a positive base tangent to $N$$N$. Since $n+2(n+1)<2m$$n+2(n+1)<2m$, by general position there is a unique point $y\in B^n\times I$$y\in B^n\times I$ such that $Fy=fx$$Fy=fx$. The tangent vector at $x$$x$ thus gives a tangent vector at $y$$y$ to $B^n\times I$$B^n\times I$. Complete this vector to a positive base tangent to $B^n\times I$$B^n\times I$, where the orientation on $B^n$$B^n$ comes from $N$$N$. The union of the images of the constructed two bases is a base at $Fy=fx$$Fy=fx$ of $\Rr^m$$\Rr^m$. If the latter base is positive, then call the initial vector 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 the latter base of $\Rr^m$$\Rr^m$, this condition indeed defines an orientation on $f\cap F$$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$. Since $m\ge n+2$$m\ge n+2$, by general position the closure $Cl\Sigma(H)$$Cl\Sigma(H)$ of the self-intersection set has codimension 2 singularities. So the closure carries a homology class with $\Zz_2$$\Zz_2$ coefficients. (Note that $Cl\Sigma(H)$$Cl\Sigma(H)$ can be assumed to be a submanifold for $2m\ge3n+2$$2m\ge3n+2$.) For $m-n$$m-n$ odd it has a natural orientation ($\S$$\S$7) and so carries a homology class with $\Zz$$\Zz$ coefficients. Define the Whitney invariant to be the homology class:

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

(The orientation is defined for each $m,n$$m,n$ but used only for odd $m-n$$m-n$. When $m-n$$m-n$ is even, for $W(f)$$W(f)$ being well-defined we need $\Zz_2$$\Zz_2$-coefficients.)

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

$\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_{\varepsilon(m-n-1)}),\quad [f] \mapsto W(f) = W(f, f_0).$ $$\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_{\varepsilon(m-n-1)}),\quad [f] \mapsto W(f) = W(f, f_0).$$

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 $H$$H$, which is proved analogously to [Skopenkov2006, $\S$$\S$2.4].

(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 $F$$F$ to be fixed on $N_0$$N_0$.) 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) Since a change of the orientation on $N$$N$ forces a change of the orientation on $B^n$$B^n$, the class $W(f)$$W(f)$ is independent of the choice of the orientation on $N$$N$. 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 we may assume that $f=f_0=\sigma\circ f$$f=f_0=\sigma\circ f$ on $N_0$$N_0$ and because a change of the orientation of $\Rr^m$$\Rr^m$ forces a change of the orientation of $f\cap F$$f\cap F$).

(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$, is $(a\mod2,0)$$(a\mod2,0)$ for $n=2$$n=2$.

(e) $W(f\#g)=W(f)$$W(f\#g)=W(f)$ for each pair of embeddings $f:N\to\Rr^m$$f:N\to\Rr^m$ and $g:S^n\to\Rr^m$$g:S^n\to\Rr^m$.

(f) For $m=2n+1$$m=2n+1$ the Whitney invariant can be recovered from the collection of pairwise linking coefficients of the components of $N$$N$, cf. Remark 3.2.b of [Skopenkov2016h].

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] 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 (Linked embedded connected sum, cf. [Skopenkov2010, Definition 1.4]). If $N$$N$ is $k$$k$-connected, then for an embedding $f_0:N\to S^{2n-k}$$f_0:N\to S^{2n-k}$ and a class $a\in H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$$a\in H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$ one can construct an embedding $f_a:N\to S^{2n-k}$$f_a:N\to S^{2n-k}$ by linked connected sum analogously to the case $k=0$$k=0$.

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$$n\ge2k+4$ in the PL or smooth categories, respectively). If $n=2k+3$$n=2k+3$, then this construction gives only a construction of embeddings $f_a:N\to\Rr^{2n-k}$$f_a:N\to\Rr^{2n-k}$ for each $a\in H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$$a\in H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$ but not a well-defined action of $H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$$H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$ on $E^{2n-k}(N)$$E^{2n-k}(N)$.

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

6.2 Classification just below the stable range

Theorem 6.2. Let $N$$N$ be a closed orientable homologically $k$$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\ge k+3$$n\ge k+3$ in the smooth category or $n\ge2k+4$$n\ge2k+4$ in the categoriy.

This was proved for $k$$k$-connected manifolds in the smooth category [Haefliger&Hirsch1963], and in the PL category in [Hudson1969, $\S$$\S$11], [Boechat&Haefliger1970], [Boechat1971], cf. [Vrabec1977]. The proof of [Haefliger&Hirsch1963], [Boechat&Haefliger1970], [Boechat1971] actually used the homological $k$$k$-connectedness assumption.

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].

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

By Theorem 6.2 the Whitney invariant $W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{\varepsilon(q)}$$W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{\varepsilon(q)}$ is bijective for $1\le p\le q-2$$1\le p\le q-2$. It is in fact a group isomorphism (for the group structure introduced in [Skopenkov2006a], [Skopenkov2015], [Skopenkov2015a]). The Hudson torus $\Hud(1)$$\Hud(1)$ generates $E^{p+2q+1}(S^p\times S^q)$$E^{p+2q+1}(S^p\times S^q)$ for $1\le p<q-1$$1\le p<q-1$; this holds by Theorem 6.2 because $W(\Hud(1),\Hud(0))=1\in\Zz_{\varepsilon(q)}$$W(\Hud(1),\Hud(0))=1\in\Zz_{\varepsilon(q)}$. Also, for $q\ge2$$q\ge2$ by Theorem 6.2 the Whitney invariants

$\displaystyle W^{3q}_{q-1,q}:E^{3q}_{PL}(S^{q-1}\times S^q)\to\Zz_{\varepsilon(q)} \quad\text{and}\quad W^{3q+1}_{q,q}:E^{3q+1}_{PL}(S^q\times S^q)\to\Zz_{\varepsilon(q)}\oplus \Zz_{\varepsilon(q)}$ $$\displaystyle W^{3q}_{q-1,q}:E^{3q}_{PL}(S^{q-1}\times S^q)\to\Zz_{\varepsilon(q)} \quad\text{and}\quad W^{3q+1}_{q,q}:E^{3q+1}_{PL}(S^q\times S^q)\to\Zz_{\varepsilon(q)}\oplus \Zz_{\varepsilon(q)}$$

are bijective. In the smooth category for $q$$q$ even $W^{3q}_{q-1,q}$$W^{3q}_{q-1,q}$ is not injective (see the next subsection), and $W^{3q+1}_{q,q}$$W^{3q+1}_{q,q}$ is not surjective [Boechat1971], [Skopenkov2016f], and $W^7_{2,2}$$W^7_{2,2}$ is not injective [Skopenkov2016f].

6.3 Classification in the presence of smoothly knotted spheres

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.

$\displaystyle E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)}$ $$\displaystyle E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)}$$

for each $s>1$$s>1$ [Haefliger1966]. The following result for $n=2k+3$$n=2k+3$ was obtained using the Boéchat-Haefliger formula for the smoothing obstruction [Boechat&Haefliger1970], [Boechat1971]. Using that formula one can define the higher-dimensional Kreck invariant [Skopenkov2008].

Theorem 6.3 [Skopenkov2008]. 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 each $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\}$.

E.g. by Theorem 6.3 the Whitney invariant $W:E^{6l}_D(S^{2l-1}\times S^{2l})\to\Zz$$W:E^{6l}_D(S^{2l-1}\times S^{2l})\to\Zz$ is surjective and for each $u\in\Zz$$u\in\Zz$ there is a 1-1 correspondence $W^{-1}u\to\Zz_u$$W^{-1}u\to\Zz_u$.

6.4 Classification further below the stable range

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 remarks on $E^{2n-1}(N)$$E^{2n-1}(N)$ in $\S$$\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]. 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

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

For $k=0$$k=0$ this is the same as General Position Theorem 2.1 [Skopenkov2016c] (because $V_{2n+1,n+1}$$V_{2n+1,n+1}$ is $(n-1)$$(n-1)$-connected). For $k=1$$k=1$ this is covered by Theorem 6.2; for $k\ge2$$k\ge2$ it is not.

E.g. by Theorem 6.4 there is a 1-1 correspondence $E^m(S^p\times S^q)\to\pi_p(V_{m,p+q+1})\oplus\pi_q(V_{m,p+q+1})$$E^m(S^p\times S^q)\to\pi_p(V_{m,p+q+1})\oplus\pi_q(V_{m,p+q+1})$ for $m\ge2q+3$$m\ge2q+3$ and $2m\ge3q+3p+4$$2m\ge3q+3p+4$. For a generalization see [Skopenkov2016k] (to knotted tori) and [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 each $M>n$$M>n$.

7 An orientation on the self-intersection set

Let $f:N\to\Rr^m$$f:N\to\Rr^m$ be a general position smooth map of an oriented $n$$n$-manifold $N$$N$. Assume that $m\ge n+2$$m\ge n+2$ so that the closure $Cl\Sigma(f)$$Cl\Sigma(f)$ of the self-intersection set of $f$$f$ has codimension 2 singularities.

Definition 7.1 (a natural orientation on $\Sigma(f)$$\Sigma(f)$). Take points $x,y\in N$$x,y\in N$ away from the singularities of $\Sigma(f)$$\Sigma(f)$ and such that $fx=fy$$fx=fy$. Then a $(2n-m)$$(2n-m)$-base $\xi_x$$\xi_x$ tangent to $\Sigma(f)$$\Sigma(f)$ at $x$$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)$$\Sigma(f)$ at $y$$y$. Since $N$$N$ is orientable, we can take positive $(m-n)$$(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$.

We remark that

• a change of the orientation of $N$$N$ forces changes of the signs of $\eta_x$$\eta_x$ and $\eta_y$$\eta_y$ and so does not change the orientation of $\Sigma(f)$$\Sigma(f)$.

• the natural orientation on $\Sigma(f)$$\Sigma(f)$ need not extend to $Cl\Sigma(f)$$Cl\Sigma(f)$: take the cone $D^3\to\Rr^5$$D^3\to\Rr^5$ over a general position map $S^2\to\Rr^4$$S^2\to\Rr^4$ having only one self-intersection point.

• the natural orientation on $\Sigma(f)$$\Sigma(f)$ extends to $Cl\Sigma(f)$$Cl\Sigma(f)$ if $m-n$$m-n$ is odd [Hudson1969, Lemma 11.4].

Definition 7.2 (a natural orientation on $f\Sigma(f)$$f\Sigma(f)$ for $m-n$$m-n$ even). Take a $(2n-m)$$(2n-m)$-base $\xi$$\xi$ at a point $x\in f\Sigma(f)$$x\in f\Sigma(f)$ away from the singularities of $f\Sigma(f)$$f\Sigma(f)$. Since $N$$N$ is orientable, we can take a positive $(m-n)$$(m-n)$-base $\eta_+$$\eta_+$ normal to $f\Sigma(f)$$f\Sigma(f)$ in one sheet of $f(N)$$f(N)$. Analogously construct an $(m-n)$$(m-n)$-base $\eta_-$$\eta_-$ for the other sheet of $f(N)$$f(N)$. Since $m-n$$m-n$ is even, the orientation of the base $(\xi,\eta_+,\eta_-)$$(\xi,\eta_+,\eta_-)$ of $\Rr^m$$\Rr^m$ does not depend on choosing the first and the other sheet of $f(N)$$f(N)$. 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)$$f\Sigma(f)$.

8 References

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\le p\le q-2$. It is in fact a group isomorphism (for the group structure introduced in \cite{Skopenkov2006a}, \cite{Skopenkov2015}, \cite{Skopenkov2015a}). The [[#Hudson tori| Hudson torus]] $\Hud(1)$ generates $E^{p+2q+1}(S^p\times S^q)$ for 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$$N \to\Rr^m$ are isotopic [Skopenkov2016c, Theorem 3.2]. In this page we summarize the situation for $m=2n\ge6$$m=2n\ge6$ and some more general situations.

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].

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, either $n$$n$ is odd or $N$$N$ is orientable, and either $n\ge4$$n\ge4$ or $n=3$$n=3$ and we are in the PL category. 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.

This is proved in [Haefliger1962b, 1.3.e], [Haefliger1963], [Haefliger&Hirsch1963, Theorem 2.4], [Bausum1975, Theorem 43] in the smooth category, and in [Weber1967], [Vrabec1977, Theorem 1.1] in the PL category.

If $n\ge4$$n\ge4$ is even and $N$$N$ is a closed connected non-orientable $n$$n$-manifold, then [Bausum1975, Theorem 43] asserts that there is a 1-1 correspondence

$\displaystyle E^{2n}(N)\to \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s$ $$\displaystyle E^{2n}(N)\to \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s$$

(because $Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$$Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$ is given by multiplication with $w_1(N)$$w_1(N)$). This 1-1 correspondence can presumably be defined as a generalized Whitney invariant, but the proof used the Haefliger-Wu invariant whose definition can be found in [Skopenkov2006, $\S$$\S$5]. It would be interesting to check if this description of $E^{2n}(N)$$E^{2n}(N)$ is equivalent to its different forms [Haefliger1962b, 1.3.e], [Haefliger1963], [Vrabec1977, Theorem 1.1].

The classification of smooth embeddings of 3-manifolds in $\Rr^6$$\Rr^6$ is more complicated, see $\S$$\S$6.3 or [Skopenkov2016t].

For embeddings of $n$$n$-manifolds in $\Rr^{2n-1}$$\Rr^{2n-1}$ see the case of 4-manifolds [Skopenkov2016f], [Yasui1984] for $n\ge5$$n\ge5$ and [Saeki1999], [Skopenkov2010], [Tonkonog2010] for non-closed manifolds.

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

3 Hudson tori

Together with the Haefliger knotted sphere [Skopenkov2016t], the examples of Hudson tori presented below were the first examples of embeddings in codimension greater than 2 which are not isotopic to the standard embedding defined below. (Hudson's construction [Hudson1963] was not as explicit as those below.)

For $m\ge n+2$$m\ge n+2$ we define the standard embedding $S^p\times S^{n-p}\to\Rr^m$$S^p\times S^{n-p}\to\Rr^m$ as the composition of the standard embeddings $S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}=\Rr^{n+2}\to\Rr^m$$S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}=\Rr^{n+2}\to\Rr^m$. (Note that in [Skopenkov2015], [Skopenkov2015a] a different embedding is called a standard embedding. Since these two embeddings are isotopic, no confusion would appear.)

Let $1_n:=(1,0,\ldots,0)\in S^n$$1_n:=(1,0,\ldots,0)\in S^n$.

Let us construct, for each $a\in\Zz$$a\in\Zz$ and $n\ge2$$n\ge2$, an embedding

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

The reader might first consider the case $n=2$$n=2$.

Definition 3.1. Take the standard embeddings $2D^{n+1}\times S^{n-1}\subset\Rr^{2n}$$2D^{n+1}\times S^{n-1}\subset\Rr^{2n}$ (where $2$$2$ means homothety with coefficient 2) and $\partial D^2\subset\partial D^{n+1}$$\partial D^2\subset\partial D^{n+1}$. Take the embedded sphere and embedded torus,

$\displaystyle 2\partial D^{n+1}\times-1_{n-1}\subset 2D^{n+1}\times S^{n-1}\subset\Rr^{2n}\quad\text{and}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$ $$\displaystyle 2\partial D^{n+1}\times-1_{n-1}\subset 2D^{n+1}\times S^{n-1}\subset\Rr^{2n}\quad\text{and}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$

Join them by an arc whose interior misses the two embedded manifolds. The Hudson torus $\Hud_n(1)$$\Hud_n(1)$ is the embedded connected sum of the two embeddings along this arc, compatible with the orientation. (Unlike the unlinked embedded connected sum [Skopenkov2016c] this is a linked embedded connected sum, i.e. the connected sum of two embeddings whose images are not contained in disjoint cubes.)

We remark that the construction in Definition 3.1 works for $n=1$$n=1$. This is not the case for the next construction in Definition 3.2.

Definition 3.2. For $a\in\Zz$$a\in\Zz$ we repeat the construction of Definition 3.1 replacing $2\partial D^{n+1}\times-1_{n-1}$$2\partial D^{n+1}\times-1_{n-1}$ by $|a|$$|a|$ copies $(1+\frac1k)\partial D^{n+1}\times-1_{n-1}$$(1+\frac1k)\partial D^{n+1}\times-1_{n-1}$ ($k=1,\dots,|a|$$k=1,\dots,|a|$) of $S^n$$S^n$. The copies are outside $D^{n+1}\times S^{n-1}$$D^{n+1}\times S^{n-1}$ and are `parallel' to $\partial D^{n+1}\times-1_{n-1}$$\partial D^{n+1}\times-1_{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 by tubes joining each $k$$k$-th copy to the $(k+1)$$(k+1)$-th copy. We obtain an embedding $g:S^n\to\Rr^{2n}$$g:S^n\to\Rr^{2n}$. Let $\Hud_n(a)$$\Hud_n(a)$ be the linked embedded connected sum of $g$$g$ with the embedding $\partial D^2\times S^{n-1}\subset\Rr^{2n}$$\partial D^2\times S^{n-1}\subset\Rr^{2n}$ from Definition 3.1.

Clearly, $\Hud_n(0)$$\Hud_n(0)$ is isotopic to the standard embedding.

The original motivation for Hudson was that $\Hud_n(1)$$\Hud_n(1)$ is not isotopic to $\Hud_n(0)$$\Hud_n(0)$ for each $n$$n$ (this is a particular case of Proposition 3.3 below).

One guesses 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 exists and is `realized' by the homotopy class of the map

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

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

Proposition 3.3. 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 3.3 follows by calculation of the Whitney invariant (Remark 5.3.e below) and, for $n$$n$ even, by Theorem 2.1.

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)$ isotopic to $\Hud_2(2)$$\Hud_2(2)$? It would also be interesting to find an explicit construction of an isotopy between $\Hud_{2k}(a)$$\Hud_{2k}(a)$ and $\Hud_{2k}(a+2)$$\Hud_{2k}(a+2)$, cf. [Vrabec1977, $\S$$\S$5].

Definition 3.4. Let us give, for $a\in\Zz$$a\in\Zz$ and $n\ge2$$n\ge2$, another construction of embeddings

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

Define a map $S^0\times S^{n-1}\to D^n$$S^0\times S^{n-1}\to D^n$ to be the constant $0\in D^n$$0\in D^n$ on one component $1\times S^{n-1}$$1\times S^{n-1}$ and the `standard inclusion' $\{-1\}\times S^{n-1}\to\partial D^n\subset D^n$$\{-1\}\times S^{n-1}\to\partial D^n\subset D^n$ on the other component. This map gives an embedding

$\displaystyle S^0\times S^{n-1}\to D^n\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$ $$\displaystyle S^0\times S^{n-1}\to D^n\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$

(See [Skopenkov2006, Figure 2.2]. The image of this embedding is the union of the standard $S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$$S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$ and the graph of the identity map in $S^{n-1}\times S^{n-1}\subset D^n\times S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$$S^{n-1}\times S^{n-1}\subset D^n\times S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$.)

Take any $x\in S^{n-1}$$x\in S^{n-1}$. The disk $D^{n+1}\times x$$D^{n+1}\times x$ intersects the image of this embedding at two points lying in $D^n\times x$$D^n\times x$, i.e., at the image of an embedding $S^0\times x\to D^n\times x$$S^0\times x\to D^n\times x$. Extend the latter embedding to an embedding $S^1\times x\to D^{n+1}\times x$$S^1\times x\to D^{n+1}\times x$. See [Skopenkov2006, Figure 2.3].) Thus we obtain the Hudson torus

$\displaystyle \Hud_n'(1):S^1\times S^{n-1}\overset h\to D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$ $$\displaystyle \Hud_n'(1):S^1\times S^{n-1}\overset h\to D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$

Here $h(e^{i\theta},y):=(y\cos\theta,\sin\theta,y)$$h(e^{i\theta},y):=(y\cos\theta,\sin\theta,y)$, where $D^{n+1}$$D^{n+1}$ is identified with $D^n\times D^1$$D^n\times D^1$.

The embedding $\Hud_n'(a)$$\Hud_n'(a)$ is obtained in the same way starting from a map $\varphi:\{-1\}\times S^{n-1}\to\partial D^n$$\varphi:\{-1\}\times S^{n-1}\to\partial D^n$ of degree $a$$a$ instead of the `standard inclusion'.

Remark 3.5. (a) The analogue of Proposition 3.3 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$$n\ge4$ and are PL isotopic for $n\ge3$$n\ge3$ [Skopenkov2006a]. It would be interesting to know if they are isotopic for $n=2$$n=2$, or if they are smoothly isotopic for $n=3$$n=3$.

(c) For $n=2$$n=2$ these construction give what we call the left Hudson torus. The right Hudson torus is constructed analogously. It is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of $S^1\times S^1$$S^1\times S^1$. The right and the left Hudson tori are not isotopic by Remark 5.3.e 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$ 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})$. There are versions of these constructions corresponding to Definition 3.4. For $p=0$$p=0$ this corresponds to the Zeeman construction [Skopenkov2016h] and its composition with the second unframed Kirby move. It would be interesting to know if the links $\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$$\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$ are isotopic, cf. [Skopenkov2015a, Remark 2.9.b]. These constructions can be further generalized [Skopenkov2016k].

4 Action by linked embedded connected sum

In this section we generalize the construction of the Hudson torus $\Hud(a)$$\Hud(a)$. For $n\ge3$$n\ge3$, a closed connected orientable $n$$n$-manifold $N$$N$, an embedding $f_0:N\to\Rr^{2n}$$f_0:N\to\Rr^{2n}$ and $a\in H_1(N;\Zz_{\varepsilon(n-1)})$$a\in H_1(N;\Zz_{\varepsilon(n-1)})$, we construct an embedding $f_a:N\to\Rr^{2n}$$f_a:N\to\Rr^{2n}$. This embedding is obtained by linked embedded connected sum of $f_0$$f_0$ with an $n$$n$-sphere representing the homology Alexander dual of $a$$a$.

More precisely, represent $a$$a$ by an embedding $a:S^1\to N$$a:S^1\to N$. 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}$. Identify $\nu_{f_0}^{-1}a(S^1)$$\nu_{f_0}^{-1}a(S^1)$ with $S^1\times S^{n-1}$$S^1\times S^{n-1}$. In the next paragraph we recall definition of embedded surgery on $S^1\times*\subset S^1\times S^{n-1}$$S^1\times*\subset S^1\times S^{n-1}$, which yields an embedding $g:S^n\to C_{f_0}$$g:S^n\to C_{f_0}$. Then we define $f_a$$f_a$ to be the (linked) embedded connected sum of $f_0$$f_0$ and $g$$g$ (along a certain arc joining their images).

Take a vector field on $S^1\times-1_{n-1}$$S^1\times-1_{n-1}$ normal to $S^1\times S^{n-1}$$S^1\times S^{n-1}$. Extend $S^1\times-1_{n-1}$$S^1\times-1_{n-1}$ along this vector field to a map $\overline a:D^2\to\Rr^{2n}$$\overline a:D^2\to\Rr^{2n}$. Since $2n>4$$2n>4$ and $n+2<2n$$n+2<2n$, by general position we may assume that $\overline a$$\overline a$ is an embedding and $\overline a(Int D^2)$$\overline a(Int D^2)$ misses $f_0(N)\cup S^1\times S^{n-1}$$f_0(N)\cup S^1\times S^{n-1}$. Since $n-1>1$$n-1>1$, we have $\pi_1(V_{2n-2,n-1})=0$$\pi_1(V_{2n-2,n-1})=0$. Hence the standard framing of $S^1\times-1_{n-1}$$S^1\times-1_{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 a(D^2)$$\overline a(D^2)$ in $\Rr^{2n}$$\Rr^{2n}$. Thus $\overline a$$\overline a$ extends to an embedding

$\displaystyle \widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}.$ $$\displaystyle \widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}.$$

Define an embedding $g:S^n\to C_{f_0}$$g:S^n\to C_{f_0}$ by setting

$\displaystyle g(S^n):\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n$ $$\displaystyle g(S^n):\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n$$

with natural orientation.

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

Proposition 4.1. Linked embedded connected sum, or parametric connected sum, 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)$, unless $n=3$$n=3$ in the smooth category.

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

5 The Whitney invariant

Let $N$$N$ be a closed $n$$n$-manifold and fix an embedding $f_0:N\to\Rr^m$$f_0:N\to\Rr^m$. For any other embedding $f \colon N \to \Rr^m$$f \colon N \to \Rr^m$, there is an invariant $W(f, f_0)$$W(f, f_0)$, called the Whitney invariant, which we define in this section. Roughly speaking, $W(f,f_0)=W_{f_0}(f)=W(f)$$W(f,f_0)=W_{f_0}(f)=W(f)$ is defined as the homology class of the self-intersection set $\Sigma(H)$$\Sigma(H)$ of a general position homotopy $H$$H$ between $f$$f$ and $f_0$$f_0$. This is formalized in Definition 5.2 in the smooth category, following [Skopenkov2010]. The definition in the PL category is analogous [Hudson1969, $\S$$\S$12], [Vrabec1977, p. 145], [Skopenkov2006, $\S$$\S$2.4]. We begin by presenting a simpler definition of the Whitney invariant, Definition 5.1, for a particular case. For Theorem 2.1 only the case $m=2n$$m=2n$ is required.

Fix an orientation on $\Rr^m$$\Rr^m$. Assume that either $m-n$$m-n$ is even or $N$$N$ is oriented.

Definition 5.1. Assume that $N$$N$ is $(2n-m)$$(2n-m)$-connected and $2m\ge3n+3$$2m\ge3n+3$. Then restrictions of $f$$f$ and $f_0$$f_0$ to $N_0$$N_0$ are regular homotopic [Hirsch1959]. 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, cf. [Haefliger&Hirsch1963, 3.1.b], [Takase2006, Lemma 2.2]. 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$$n+2(n+1)<2m$, by general position $Cl(f\cap F)$$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. For $m-n$$m-n$ odd it has a natural orientation defined below, and so carries a homology class with $\Zz$$\Zz$ coefficients. Define $W(f)$$W(f)$ to be the homology class:

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

The orientation on $f\cap F$$f\cap F$ (extendable to $Cl(f\cap F)$$Cl(f\cap F)$) is defined for $m-n$$m-n$ odd as follows. For each point $x\in f\cap F$$x\in f\cap F$ take a vector at $x$$x$ tangent to $f\cap F$$f\cap F$. Complete this vector to a positive base tangent to $N$$N$. Since $n+2(n+1)<2m$$n+2(n+1)<2m$, by general position there is a unique point $y\in B^n\times I$$y\in B^n\times I$ such that $Fy=fx$$Fy=fx$. The tangent vector at $x$$x$ thus gives a tangent vector at $y$$y$ to $B^n\times I$$B^n\times I$. Complete this vector to a positive base tangent to $B^n\times I$$B^n\times I$, where the orientation on $B^n$$B^n$ comes from $N$$N$. The union of the images of the constructed two bases is a base at $Fy=fx$$Fy=fx$ of $\Rr^m$$\Rr^m$. If the latter base is positive, then call the initial vector 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 the latter base of $\Rr^m$$\Rr^m$, this condition indeed defines an orientation on $f\cap F$$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$. Since $m\ge n+2$$m\ge n+2$, by general position the closure $Cl\Sigma(H)$$Cl\Sigma(H)$ of the self-intersection set has codimension 2 singularities. So the closure carries a homology class with $\Zz_2$$\Zz_2$ coefficients. (Note that $Cl\Sigma(H)$$Cl\Sigma(H)$ can be assumed to be a submanifold for $2m\ge3n+2$$2m\ge3n+2$.) For $m-n$$m-n$ odd it has a natural orientation ($\S$$\S$7) and so carries a homology class with $\Zz$$\Zz$ coefficients. Define the Whitney invariant to be the homology class:

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

(The orientation is defined for each $m,n$$m,n$ but used only for odd $m-n$$m-n$. When $m-n$$m-n$ is even, for $W(f)$$W(f)$ being well-defined we need $\Zz_2$$\Zz_2$-coefficients.)

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

$\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_{\varepsilon(m-n-1)}),\quad [f] \mapsto W(f) = W(f, f_0).$ $$\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_{\varepsilon(m-n-1)}),\quad [f] \mapsto W(f) = W(f, f_0).$$

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 $H$$H$, which is proved analogously to [Skopenkov2006, $\S$$\S$2.4].

(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 $F$$F$ to be fixed on $N_0$$N_0$.) 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) Since a change of the orientation on $N$$N$ forces a change of the orientation on $B^n$$B^n$, the class $W(f)$$W(f)$ is independent of the choice of the orientation on $N$$N$. 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 we may assume that $f=f_0=\sigma\circ f$$f=f_0=\sigma\circ f$ on $N_0$$N_0$ and because a change of the orientation of $\Rr^m$$\Rr^m$ forces a change of the orientation of $f\cap F$$f\cap F$).

(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$, is $(a\mod2,0)$$(a\mod2,0)$ for $n=2$$n=2$.

(e) $W(f\#g)=W(f)$$W(f\#g)=W(f)$ for each pair of embeddings $f:N\to\Rr^m$$f:N\to\Rr^m$ and $g:S^n\to\Rr^m$$g:S^n\to\Rr^m$.

(f) For $m=2n+1$$m=2n+1$ the Whitney invariant can be recovered from the collection of pairwise linking coefficients of the components of $N$$N$, cf. Remark 3.2.b of [Skopenkov2016h].

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] 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 (Linked embedded connected sum, cf. [Skopenkov2010, Definition 1.4]). If $N$$N$ is $k$$k$-connected, then for an embedding $f_0:N\to S^{2n-k}$$f_0:N\to S^{2n-k}$ and a class $a\in H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$$a\in H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$ one can construct an embedding $f_a:N\to S^{2n-k}$$f_a:N\to S^{2n-k}$ by linked connected sum analogously to the case $k=0$$k=0$.

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$$n\ge2k+4$ in the PL or smooth categories, respectively). If $n=2k+3$$n=2k+3$, then this construction gives only a construction of embeddings $f_a:N\to\Rr^{2n-k}$$f_a:N\to\Rr^{2n-k}$ for each $a\in H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$$a\in H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$ but not a well-defined action of $H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$$H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$ on $E^{2n-k}(N)$$E^{2n-k}(N)$.

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

6.2 Classification just below the stable range

Theorem 6.2. Let $N$$N$ be a closed orientable homologically $k$$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\ge k+3$$n\ge k+3$ in the smooth category or $n\ge2k+4$$n\ge2k+4$ in the categoriy.

This was proved for $k$$k$-connected manifolds in the smooth category [Haefliger&Hirsch1963], and in the PL category in [Hudson1969, $\S$$\S$11], [Boechat&Haefliger1970], [Boechat1971], cf. [Vrabec1977]. The proof of [Haefliger&Hirsch1963], [Boechat&Haefliger1970], [Boechat1971] actually used the homological $k$$k$-connectedness assumption.

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].

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

By Theorem 6.2 the Whitney invariant $W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{\varepsilon(q)}$$W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{\varepsilon(q)}$ is bijective for $1\le p\le q-2$$1\le p\le q-2$. It is in fact a group isomorphism (for the group structure introduced in [Skopenkov2006a], [Skopenkov2015], [Skopenkov2015a]). The Hudson torus $\Hud(1)$$\Hud(1)$ generates $E^{p+2q+1}(S^p\times S^q)$$E^{p+2q+1}(S^p\times S^q)$ for $1\le p<q-1$$1\le p<q-1$; this holds by Theorem 6.2 because $W(\Hud(1),\Hud(0))=1\in\Zz_{\varepsilon(q)}$$W(\Hud(1),\Hud(0))=1\in\Zz_{\varepsilon(q)}$. Also, for $q\ge2$$q\ge2$ by Theorem 6.2 the Whitney invariants

$\displaystyle W^{3q}_{q-1,q}:E^{3q}_{PL}(S^{q-1}\times S^q)\to\Zz_{\varepsilon(q)} \quad\text{and}\quad W^{3q+1}_{q,q}:E^{3q+1}_{PL}(S^q\times S^q)\to\Zz_{\varepsilon(q)}\oplus \Zz_{\varepsilon(q)}$ $$\displaystyle W^{3q}_{q-1,q}:E^{3q}_{PL}(S^{q-1}\times S^q)\to\Zz_{\varepsilon(q)} \quad\text{and}\quad W^{3q+1}_{q,q}:E^{3q+1}_{PL}(S^q\times S^q)\to\Zz_{\varepsilon(q)}\oplus \Zz_{\varepsilon(q)}$$

are bijective. In the smooth category for $q$$q$ even $W^{3q}_{q-1,q}$$W^{3q}_{q-1,q}$ is not injective (see the next subsection), and $W^{3q+1}_{q,q}$$W^{3q+1}_{q,q}$ is not surjective [Boechat1971], [Skopenkov2016f], and $W^7_{2,2}$$W^7_{2,2}$ is not injective [Skopenkov2016f].

6.3 Classification in the presence of smoothly knotted spheres

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.

$\displaystyle E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)}$ $$\displaystyle E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)}$$

for each $s>1$$s>1$ [Haefliger1966]. The following result for $n=2k+3$$n=2k+3$ was obtained using the Boéchat-Haefliger formula for the smoothing obstruction [Boechat&Haefliger1970], [Boechat1971]. Using that formula one can define the higher-dimensional Kreck invariant [Skopenkov2008].

Theorem 6.3 [Skopenkov2008]. 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 each $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\}$.

E.g. by Theorem 6.3 the Whitney invariant $W:E^{6l}_D(S^{2l-1}\times S^{2l})\to\Zz$$W:E^{6l}_D(S^{2l-1}\times S^{2l})\to\Zz$ is surjective and for each $u\in\Zz$$u\in\Zz$ there is a 1-1 correspondence $W^{-1}u\to\Zz_u$$W^{-1}u\to\Zz_u$.

6.4 Classification further below the stable range

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 remarks on $E^{2n-1}(N)$$E^{2n-1}(N)$ in $\S$$\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]. 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

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

For $k=0$$k=0$ this is the same as General Position Theorem 2.1 [Skopenkov2016c] (because $V_{2n+1,n+1}$$V_{2n+1,n+1}$ is $(n-1)$$(n-1)$-connected). For $k=1$$k=1$ this is covered by Theorem 6.2; for $k\ge2$$k\ge2$ it is not.

E.g. by Theorem 6.4 there is a 1-1 correspondence $E^m(S^p\times S^q)\to\pi_p(V_{m,p+q+1})\oplus\pi_q(V_{m,p+q+1})$$E^m(S^p\times S^q)\to\pi_p(V_{m,p+q+1})\oplus\pi_q(V_{m,p+q+1})$ for $m\ge2q+3$$m\ge2q+3$ and $2m\ge3q+3p+4$$2m\ge3q+3p+4$. For a generalization see [Skopenkov2016k] (to knotted tori) and [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 each $M>n$$M>n$.

7 An orientation on the self-intersection set

Let $f:N\to\Rr^m$$f:N\to\Rr^m$ be a general position smooth map of an oriented $n$$n$-manifold $N$$N$. Assume that $m\ge n+2$$m\ge n+2$ so that the closure $Cl\Sigma(f)$$Cl\Sigma(f)$ of the self-intersection set of $f$$f$ has codimension 2 singularities.

Definition 7.1 (a natural orientation on $\Sigma(f)$$\Sigma(f)$). Take points $x,y\in N$$x,y\in N$ away from the singularities of $\Sigma(f)$$\Sigma(f)$ and such that $fx=fy$$fx=fy$. Then a $(2n-m)$$(2n-m)$-base $\xi_x$$\xi_x$ tangent to $\Sigma(f)$$\Sigma(f)$ at $x$$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)$$\Sigma(f)$ at $y$$y$. Since $N$$N$ is orientable, we can take positive $(m-n)$$(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$.

We remark that

• a change of the orientation of $N$$N$ forces changes of the signs of $\eta_x$$\eta_x$ and $\eta_y$$\eta_y$ and so does not change the orientation of $\Sigma(f)$$\Sigma(f)$.

• the natural orientation on $\Sigma(f)$$\Sigma(f)$ need not extend to $Cl\Sigma(f)$$Cl\Sigma(f)$: take the cone $D^3\to\Rr^5$$D^3\to\Rr^5$ over a general position map $S^2\to\Rr^4$$S^2\to\Rr^4$ having only one self-intersection point.

• the natural orientation on $\Sigma(f)$$\Sigma(f)$ extends to $Cl\Sigma(f)$$Cl\Sigma(f)$ if $m-n$$m-n$ is odd [Hudson1969, Lemma 11.4].

Definition 7.2 (a natural orientation on $f\Sigma(f)$$f\Sigma(f)$ for $m-n$$m-n$ even). Take a $(2n-m)$$(2n-m)$-base $\xi$$\xi$ at a point $x\in f\Sigma(f)$$x\in f\Sigma(f)$ away from the singularities of $f\Sigma(f)$$f\Sigma(f)$. Since $N$$N$ is orientable, we can take a positive $(m-n)$$(m-n)$-base $\eta_+$$\eta_+$ normal to $f\Sigma(f)$$f\Sigma(f)$ in one sheet of $f(N)$$f(N)$. Analogously construct an $(m-n)$$(m-n)$-base $\eta_-$$\eta_-$ for the other sheet of $f(N)$$f(N)$. Since $m-n$$m-n$ is even, the orientation of the base $(\xi,\eta_+,\eta_-)$$(\xi,\eta_+,\eta_-)$ of $\Rr^m$$\Rr^m$ does not depend on choosing the first and the other sheet of $f(N)$$f(N)$. 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)$$f\Sigma(f)$.

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\le p === Classification in the presence of smoothly knotted spheres === ; \label{s:knotted} 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 each $s>1$ \cite{Haefliger1966}. The following result for $n=2k+3$ was obtained using the Boéchat-Haefliger formula for the smoothing obstruction \cite{Boechat&Haefliger1970}, \cite{Boechat1971}. Using that formula one can define the higher-dimensional [[3-manifolds in 6-space#The Kreck invariant|Kreck invariant]] \cite{Skopenkov2008}. {{beginthm|Theorem|\cite{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 each $u\in H_{2l-1}(N)$ 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 $\max\{d\in\Zz\ | \ \text{there is }x_1\in G: \ x=dx_1\}$. E.g. by Theorem \ref{hicod} the Whitney invariant $W:E^{6l}_D(S^{2l-1}\times S^{2l})\to\Zz$ is surjective and for each $u\in\Zz$ there is a 1-1 correspondence $W^{-1}u\to\Zz_u$. === Classification further below the stable range === ; How does one describe $E^m(N)$ when $N$ is not $(2n-m)$-connected? For general $N$ see the [[#Classification|remarks]] on $E^{2n-1}(N)$ in $\S. 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{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}} For $k=0$ this is the same as [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting_theorems|General Position Theorem 2.1]] \cite{Skopenkov2016c} (because $V_{2n+1,n+1}$ is $(n-1)$-connected). For $k=1$ this is covered by Theorem \ref{hico}; for $k\ge2$ it is not. E.g. by Theorem \ref{begl} there is a 1-1 correspondence $E^m(S^p\times S^q)\to\pi_p(V_{m,p+q+1})\oplus\pi_q(V_{m,p+q+1})$ for $m\ge2q+3$ and m\ge3q+3p+4$. For a generalization see \cite{Skopenkov2016k} (to [[Knotted tori|knotted tori]]) and \cite{Skopenkov2002}. Observe that in Theorem \ref{begl} $V_{m,n+1}$ can be replaced by $V_{M,M+n-m+1}$ for each $M>n$. == An orientation on the self-intersection set == ; Let $f:N\to\Rr^m$ be a general position smooth map of an oriented $n$-manifold $N$. Assume that $m\ge n+2$ so that the closure $Cl\Sigma(f)$ of the self-intersection set of $f$ has codimension 2 singularities. {{beginthm|Definition|(a natural orientation on $\Sigma(f)$)}}\label{oris} Take points $x,y\in N$ away from the singularities of $\Sigma(f)$ and such that $fx=fy$. Then a $(2n-m)$-base $\xi_x$ tangent to $\Sigma(f)$ at $x$ gives a $(2n-m)$-base $\xi_y:=df_y^{-1}df_x(\xi_x)$ tangent to $\Sigma(f)$ at $y$. Since $N$ is orientable, 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}} We remark that * 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 $\Sigma(f)$. * the natural orientation on $\Sigma(f)$ need not extend to $Cl\Sigma(f)$: take the cone $D^3\to\Rr^5$ over a general position map $S^2\to\Rr^4$ having only one self-intersection point. * the natural orientation on $\Sigma(f)$ extends to $Cl\Sigma(f)$ if $m-n$ is odd \cite[Lemma 11.4]{Hudson1969}. {{beginthm|Definition|(a natural orientation on $f\Sigma(f)$ for $m-n$ even)}}\label{orifs} Take a $(2n-m)$-base $\xi$ at a point $x\in f\Sigma(f)$ away from the singularities of $f\Sigma(f)$. Since $N$ is orientable, we can take a positive $(m-n)$-base $\eta_+$ normal to $f\Sigma(f)$ in one sheet of $f(N)$. Analogously construct an $(m-n)$-base $\eta_-$ for the other sheet of $f(N)$. 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 $f(N)$. 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 $(\xi,\eta_+,\eta_-)$. {{endthm}} 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 $f\Sigma(f)$. == 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$$N \to\Rr^m$ are isotopic [Skopenkov2016c, Theorem 3.2]. In this page we summarize the situation for $m=2n\ge6$$m=2n\ge6$ and some more general situations.

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].

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, either $n$$n$ is odd or $N$$N$ is orientable, and either $n\ge4$$n\ge4$ or $n=3$$n=3$ and we are in the PL category. 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.

This is proved in [Haefliger1962b, 1.3.e], [Haefliger1963], [Haefliger&Hirsch1963, Theorem 2.4], [Bausum1975, Theorem 43] in the smooth category, and in [Weber1967], [Vrabec1977, Theorem 1.1] in the PL category.

If $n\ge4$$n\ge4$ is even and $N$$N$ is a closed connected non-orientable $n$$n$-manifold, then [Bausum1975, Theorem 43] asserts that there is a 1-1 correspondence

$\displaystyle E^{2n}(N)\to \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s$ $$\displaystyle E^{2n}(N)\to \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s$$

(because $Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$$Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$ is given by multiplication with $w_1(N)$$w_1(N)$). This 1-1 correspondence can presumably be defined as a generalized Whitney invariant, but the proof used the Haefliger-Wu invariant whose definition can be found in [Skopenkov2006, $\S$$\S$5]. It would be interesting to check if this description of $E^{2n}(N)$$E^{2n}(N)$ is equivalent to its different forms [Haefliger1962b, 1.3.e], [Haefliger1963], [Vrabec1977, Theorem 1.1].

The classification of smooth embeddings of 3-manifolds in $\Rr^6$$\Rr^6$ is more complicated, see $\S$$\S$6.3 or [Skopenkov2016t].

For embeddings of $n$$n$-manifolds in $\Rr^{2n-1}$$\Rr^{2n-1}$ see the case of 4-manifolds [Skopenkov2016f], [Yasui1984] for $n\ge5$$n\ge5$ and [Saeki1999], [Skopenkov2010], [Tonkonog2010] for non-closed manifolds.

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

3 Hudson tori

Together with the Haefliger knotted sphere [Skopenkov2016t], the examples of Hudson tori presented below were the first examples of embeddings in codimension greater than 2 which are not isotopic to the standard embedding defined below. (Hudson's construction [Hudson1963] was not as explicit as those below.)

For $m\ge n+2$$m\ge n+2$ we define the standard embedding $S^p\times S^{n-p}\to\Rr^m$$S^p\times S^{n-p}\to\Rr^m$ as the composition of the standard embeddings $S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}=\Rr^{n+2}\to\Rr^m$$S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}=\Rr^{n+2}\to\Rr^m$. (Note that in [Skopenkov2015], [Skopenkov2015a] a different embedding is called a standard embedding. Since these two embeddings are isotopic, no confusion would appear.)

Let $1_n:=(1,0,\ldots,0)\in S^n$$1_n:=(1,0,\ldots,0)\in S^n$.

Let us construct, for each $a\in\Zz$$a\in\Zz$ and $n\ge2$$n\ge2$, an embedding

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

The reader might first consider the case $n=2$$n=2$.

Definition 3.1. Take the standard embeddings $2D^{n+1}\times S^{n-1}\subset\Rr^{2n}$$2D^{n+1}\times S^{n-1}\subset\Rr^{2n}$ (where $2$$2$ means homothety with coefficient 2) and $\partial D^2\subset\partial D^{n+1}$$\partial D^2\subset\partial D^{n+1}$. Take the embedded sphere and embedded torus,

$\displaystyle 2\partial D^{n+1}\times-1_{n-1}\subset 2D^{n+1}\times S^{n-1}\subset\Rr^{2n}\quad\text{and}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$ $$\displaystyle 2\partial D^{n+1}\times-1_{n-1}\subset 2D^{n+1}\times S^{n-1}\subset\Rr^{2n}\quad\text{and}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$

Join them by an arc whose interior misses the two embedded manifolds. The Hudson torus $\Hud_n(1)$$\Hud_n(1)$ is the embedded connected sum of the two embeddings along this arc, compatible with the orientation. (Unlike the unlinked embedded connected sum [Skopenkov2016c] this is a linked embedded connected sum, i.e. the connected sum of two embeddings whose images are not contained in disjoint cubes.)

We remark that the construction in Definition 3.1 works for $n=1$$n=1$. This is not the case for the next construction in Definition 3.2.

Definition 3.2. For $a\in\Zz$$a\in\Zz$ we repeat the construction of Definition 3.1 replacing $2\partial D^{n+1}\times-1_{n-1}$$2\partial D^{n+1}\times-1_{n-1}$ by $|a|$$|a|$ copies $(1+\frac1k)\partial D^{n+1}\times-1_{n-1}$$(1+\frac1k)\partial D^{n+1}\times-1_{n-1}$ ($k=1,\dots,|a|$$k=1,\dots,|a|$) of $S^n$$S^n$. The copies are outside $D^{n+1}\times S^{n-1}$$D^{n+1}\times S^{n-1}$ and are `parallel' to $\partial D^{n+1}\times-1_{n-1}$$\partial D^{n+1}\times-1_{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 by tubes joining each $k$$k$-th copy to the $(k+1)$$(k+1)$-th copy. We obtain an embedding $g:S^n\to\Rr^{2n}$$g:S^n\to\Rr^{2n}$. Let $\Hud_n(a)$$\Hud_n(a)$ be the linked embedded connected sum of $g$$g$ with the embedding $\partial D^2\times S^{n-1}\subset\Rr^{2n}$$\partial D^2\times S^{n-1}\subset\Rr^{2n}$ from Definition 3.1.

Clearly, $\Hud_n(0)$$\Hud_n(0)$ is isotopic to the standard embedding.

The original motivation for Hudson was that $\Hud_n(1)$$\Hud_n(1)$ is not isotopic to $\Hud_n(0)$$\Hud_n(0)$ for each $n$$n$ (this is a particular case of Proposition 3.3 below).

One guesses 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 exists and is `realized' by the homotopy class of the map

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

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

Proposition 3.3. 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 3.3 follows by calculation of the Whitney invariant (Remark 5.3.e below) and, for $n$$n$ even, by Theorem 2.1.

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)$ isotopic to $\Hud_2(2)$$\Hud_2(2)$? It would also be interesting to find an explicit construction of an isotopy between $\Hud_{2k}(a)$$\Hud_{2k}(a)$ and $\Hud_{2k}(a+2)$$\Hud_{2k}(a+2)$, cf. [Vrabec1977, $\S$$\S$5].

Definition 3.4. Let us give, for $a\in\Zz$$a\in\Zz$ and $n\ge2$$n\ge2$, another construction of embeddings

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

Define a map $S^0\times S^{n-1}\to D^n$$S^0\times S^{n-1}\to D^n$ to be the constant $0\in D^n$$0\in D^n$ on one component $1\times S^{n-1}$$1\times S^{n-1}$ and the `standard inclusion' $\{-1\}\times S^{n-1}\to\partial D^n\subset D^n$$\{-1\}\times S^{n-1}\to\partial D^n\subset D^n$ on the other component. This map gives an embedding

$\displaystyle S^0\times S^{n-1}\to D^n\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$ $$\displaystyle S^0\times S^{n-1}\to D^n\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$

(See [Skopenkov2006, Figure 2.2]. The image of this embedding is the union of the standard $S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$$S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$ and the graph of the identity map in $S^{n-1}\times S^{n-1}\subset D^n\times S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$$S^{n-1}\times S^{n-1}\subset D^n\times S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$.)

Take any $x\in S^{n-1}$$x\in S^{n-1}$. The disk $D^{n+1}\times x$$D^{n+1}\times x$ intersects the image of this embedding at two points lying in $D^n\times x$$D^n\times x$, i.e., at the image of an embedding $S^0\times x\to D^n\times x$$S^0\times x\to D^n\times x$. Extend the latter embedding to an embedding $S^1\times x\to D^{n+1}\times x$$S^1\times x\to D^{n+1}\times x$. See [Skopenkov2006, Figure 2.3].) Thus we obtain the Hudson torus

$\displaystyle \Hud_n'(1):S^1\times S^{n-1}\overset h\to D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$ $$\displaystyle \Hud_n'(1):S^1\times S^{n-1}\overset h\to D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$

Here $h(e^{i\theta},y):=(y\cos\theta,\sin\theta,y)$$h(e^{i\theta},y):=(y\cos\theta,\sin\theta,y)$, where $D^{n+1}$$D^{n+1}$ is identified with $D^n\times D^1$$D^n\times D^1$.

The embedding $\Hud_n'(a)$$\Hud_n'(a)$ is obtained in the same way starting from a map $\varphi:\{-1\}\times S^{n-1}\to\partial D^n$$\varphi:\{-1\}\times S^{n-1}\to\partial D^n$ of degree $a$$a$ instead of the `standard inclusion'.

Remark 3.5. (a) The analogue of Proposition 3.3 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$$n\ge4$ and are PL isotopic for $n\ge3$$n\ge3$ [Skopenkov2006a]. It would be interesting to know if they are isotopic for $n=2$$n=2$, or if they are smoothly isotopic for $n=3$$n=3$.

(c) For $n=2$$n=2$ these construction give what we call the left Hudson torus. The right Hudson torus is constructed analogously. It is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of $S^1\times S^1$$S^1\times S^1$. The right and the left Hudson tori are not isotopic by Remark 5.3.e 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$ 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})$. There are versions of these constructions corresponding to Definition 3.4. For $p=0$$p=0$ this corresponds to the Zeeman construction [Skopenkov2016h] and its composition with the second unframed Kirby move. It would be interesting to know if the links $\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$$\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$ are isotopic, cf. [Skopenkov2015a, Remark 2.9.b]. These constructions can be further generalized [Skopenkov2016k].

4 Action by linked embedded connected sum

In this section we generalize the construction of the Hudson torus $\Hud(a)$$\Hud(a)$. For $n\ge3$$n\ge3$, a closed connected orientable $n$$n$-manifold $N$$N$, an embedding $f_0:N\to\Rr^{2n}$$f_0:N\to\Rr^{2n}$ and $a\in H_1(N;\Zz_{\varepsilon(n-1)})$$a\in H_1(N;\Zz_{\varepsilon(n-1)})$, we construct an embedding $f_a:N\to\Rr^{2n}$$f_a:N\to\Rr^{2n}$. This embedding is obtained by linked embedded connected sum of $f_0$$f_0$ with an $n$$n$-sphere representing the homology Alexander dual of $a$$a$.

More precisely, represent $a$$a$ by an embedding $a:S^1\to N$$a:S^1\to N$. 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}$. Identify $\nu_{f_0}^{-1}a(S^1)$$\nu_{f_0}^{-1}a(S^1)$ with $S^1\times S^{n-1}$$S^1\times S^{n-1}$. In the next paragraph we recall definition of embedded surgery on $S^1\times*\subset S^1\times S^{n-1}$$S^1\times*\subset S^1\times S^{n-1}$, which yields an embedding $g:S^n\to C_{f_0}$$g:S^n\to C_{f_0}$. Then we define $f_a$$f_a$ to be the (linked) embedded connected sum of $f_0$$f_0$ and $g$$g$ (along a certain arc joining their images).

Take a vector field on $S^1\times-1_{n-1}$$S^1\times-1_{n-1}$ normal to $S^1\times S^{n-1}$$S^1\times S^{n-1}$. Extend $S^1\times-1_{n-1}$$S^1\times-1_{n-1}$ along this vector field to a map $\overline a:D^2\to\Rr^{2n}$$\overline a:D^2\to\Rr^{2n}$. Since $2n>4$$2n>4$ and $n+2<2n$$n+2<2n$, by general position we may assume that $\overline a$$\overline a$ is an embedding and $\overline a(Int D^2)$$\overline a(Int D^2)$ misses $f_0(N)\cup S^1\times S^{n-1}$$f_0(N)\cup S^1\times S^{n-1}$. Since $n-1>1$$n-1>1$, we have $\pi_1(V_{2n-2,n-1})=0$$\pi_1(V_{2n-2,n-1})=0$. Hence the standard framing of $S^1\times-1_{n-1}$$S^1\times-1_{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 a(D^2)$$\overline a(D^2)$ in $\Rr^{2n}$$\Rr^{2n}$. Thus $\overline a$$\overline a$ extends to an embedding

$\displaystyle \widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}.$ $$\displaystyle \widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}.$$

Define an embedding $g:S^n\to C_{f_0}$$g:S^n\to C_{f_0}$ by setting

$\displaystyle g(S^n):\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n$ $$\displaystyle g(S^n):\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n$$

with natural orientation.

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

Proposition 4.1. Linked embedded connected sum, or parametric connected sum, 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)$, unless $n=3$$n=3$ in the smooth category.

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

5 The Whitney invariant

Let $N$$N$ be a closed $n$$n$-manifold and fix an embedding $f_0:N\to\Rr^m$$f_0:N\to\Rr^m$. For any other embedding $f \colon N \to \Rr^m$$f \colon N \to \Rr^m$, there is an invariant $W(f, f_0)$$W(f, f_0)$, called the Whitney invariant, which we define in this section. Roughly speaking, $W(f,f_0)=W_{f_0}(f)=W(f)$$W(f,f_0)=W_{f_0}(f)=W(f)$ is defined as the homology class of the self-intersection set $\Sigma(H)$$\Sigma(H)$ of a general position homotopy $H$$H$ between $f$$f$ and $f_0$$f_0$. This is formalized in Definition 5.2 in the smooth category, following [Skopenkov2010]. The definition in the PL category is analogous [Hudson1969, $\S$$\S$12], [Vrabec1977, p. 145], [Skopenkov2006, $\S$$\S$2.4]. We begin by presenting a simpler definition of the Whitney invariant, Definition 5.1, for a particular case. For Theorem 2.1 only the case $m=2n$$m=2n$ is required.

Fix an orientation on $\Rr^m$$\Rr^m$. Assume that either $m-n$$m-n$ is even or $N$$N$ is oriented.

Definition 5.1. Assume that $N$$N$ is $(2n-m)$$(2n-m)$-connected and $2m\ge3n+3$$2m\ge3n+3$. Then restrictions of $f$$f$ and $f_0$$f_0$ to $N_0$$N_0$ are regular homotopic [Hirsch1959]. 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, cf. [Haefliger&Hirsch1963, 3.1.b], [Takase2006, Lemma 2.2]. 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$$n+2(n+1)<2m$, by general position $Cl(f\cap F)$$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. For $m-n$$m-n$ odd it has a natural orientation defined below, and so carries a homology class with $\Zz$$\Zz$ coefficients. Define $W(f)$$W(f)$ to be the homology class:

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

The orientation on $f\cap F$$f\cap F$ (extendable to $Cl(f\cap F)$$Cl(f\cap F)$) is defined for $m-n$$m-n$ odd as follows. For each point $x\in f\cap F$$x\in f\cap F$ take a vector at $x$$x$ tangent to $f\cap F$$f\cap F$. Complete this vector to a positive base tangent to $N$$N$. Since $n+2(n+1)<2m$$n+2(n+1)<2m$, by general position there is a unique point $y\in B^n\times I$$y\in B^n\times I$ such that $Fy=fx$$Fy=fx$. The tangent vector at $x$$x$ thus gives a tangent vector at $y$$y$ to $B^n\times I$$B^n\times I$. Complete this vector to a positive base tangent to $B^n\times I$$B^n\times I$, where the orientation on $B^n$$B^n$ comes from $N$$N$. The union of the images of the constructed two bases is a base at $Fy=fx$$Fy=fx$ of $\Rr^m$$\Rr^m$. If the latter base is positive, then call the initial vector 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 the latter base of $\Rr^m$$\Rr^m$, this condition indeed defines an orientation on $f\cap F$$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$. Since $m\ge n+2$$m\ge n+2$, by general position the closure $Cl\Sigma(H)$$Cl\Sigma(H)$ of the self-intersection set has codimension 2 singularities. So the closure carries a homology class with $\Zz_2$$\Zz_2$ coefficients. (Note that $Cl\Sigma(H)$$Cl\Sigma(H)$ can be assumed to be a submanifold for $2m\ge3n+2$$2m\ge3n+2$.) For $m-n$$m-n$ odd it has a natural orientation ($\S$$\S$7) and so carries a homology class with $\Zz$$\Zz$ coefficients. Define the Whitney invariant to be the homology class:

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

(The orientation is defined for each $m,n$$m,n$ but used only for odd $m-n$$m-n$. When $m-n$$m-n$ is even, for $W(f)$$W(f)$ being well-defined we need $\Zz_2$$\Zz_2$-coefficients.)

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

$\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_{\varepsilon(m-n-1)}),\quad [f] \mapsto W(f) = W(f, f_0).$ $$\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_{\varepsilon(m-n-1)}),\quad [f] \mapsto W(f) = W(f, f_0).$$

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 $H$$H$, which is proved analogously to [Skopenkov2006, $\S$$\S$2.4].

(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 $F$$F$ to be fixed on $N_0$$N_0$.) 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) Since a change of the orientation on $N$$N$ forces a change of the orientation on $B^n$$B^n$, the class $W(f)$$W(f)$ is independent of the choice of the orientation on $N$$N$. 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 we may assume that $f=f_0=\sigma\circ f$$f=f_0=\sigma\circ f$ on $N_0$$N_0$ and because a change of the orientation of $\Rr^m$$\Rr^m$ forces a change of the orientation of $f\cap F$$f\cap F$).

(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$, is $(a\mod2,0)$$(a\mod2,0)$ for $n=2$$n=2$.

(e) $W(f\#g)=W(f)$$W(f\#g)=W(f)$ for each pair of embeddings $f:N\to\Rr^m$$f:N\to\Rr^m$ and $g:S^n\to\Rr^m$$g:S^n\to\Rr^m$.

(f) For $m=2n+1$$m=2n+1$ the Whitney invariant can be recovered from the collection of pairwise linking coefficients of the components of $N$$N$, cf. Remark 3.2.b of [Skopenkov2016h].

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] 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 (Linked embedded connected sum, cf. [Skopenkov2010, Definition 1.4]). If $N$$N$ is $k$$k$-connected, then for an embedding $f_0:N\to S^{2n-k}$$f_0:N\to S^{2n-k}$ and a class $a\in H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$$a\in H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$ one can construct an embedding $f_a:N\to S^{2n-k}$$f_a:N\to S^{2n-k}$ by linked connected sum analogously to the case $k=0$$k=0$.

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$$n\ge2k+4$ in the PL or smooth categories, respectively). If $n=2k+3$$n=2k+3$, then this construction gives only a construction of embeddings $f_a:N\to\Rr^{2n-k}$$f_a:N\to\Rr^{2n-k}$ for each $a\in H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$$a\in H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$ but not a well-defined action of $H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$$H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$ on $E^{2n-k}(N)$$E^{2n-k}(N)$.

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

6.2 Classification just below the stable range

Theorem 6.2. Let $N$$N$ be a closed orientable homologically $k$$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\ge k+3$$n\ge k+3$ in the smooth category or $n\ge2k+4$$n\ge2k+4$ in the categoriy.

This was proved for $k$$k$-connected manifolds in the smooth category [Haefliger&Hirsch1963], and in the PL category in [Hudson1969, $\S$$\S$11], [Boechat&Haefliger1970], [Boechat1971], cf. [Vrabec1977]. The proof of [Haefliger&Hirsch1963], [Boechat&Haefliger1970], [Boechat1971] actually used the homological $k$$k$-connectedness assumption.

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].

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

By Theorem 6.2 the Whitney invariant $W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{\varepsilon(q)}$$W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{\varepsilon(q)}$ is bijective for $1\le p\le q-2$$1\le p\le q-2$. It is in fact a group isomorphism (for the group structure introduced in [Skopenkov2006a], [Skopenkov2015], [Skopenkov2015a]). The Hudson torus $\Hud(1)$$\Hud(1)$ generates $E^{p+2q+1}(S^p\times S^q)$$E^{p+2q+1}(S^p\times S^q)$ for $1\le p<q-1$$1\le p<q-1$; this holds by Theorem 6.2 because $W(\Hud(1),\Hud(0))=1\in\Zz_{\varepsilon(q)}$$W(\Hud(1),\Hud(0))=1\in\Zz_{\varepsilon(q)}$. Also, for $q\ge2$$q\ge2$ by Theorem 6.2 the Whitney invariants

$\displaystyle W^{3q}_{q-1,q}:E^{3q}_{PL}(S^{q-1}\times S^q)\to\Zz_{\varepsilon(q)} \quad\text{and}\quad W^{3q+1}_{q,q}:E^{3q+1}_{PL}(S^q\times S^q)\to\Zz_{\varepsilon(q)}\oplus \Zz_{\varepsilon(q)}$ $$\displaystyle W^{3q}_{q-1,q}:E^{3q}_{PL}(S^{q-1}\times S^q)\to\Zz_{\varepsilon(q)} \quad\text{and}\quad W^{3q+1}_{q,q}:E^{3q+1}_{PL}(S^q\times S^q)\to\Zz_{\varepsilon(q)}\oplus \Zz_{\varepsilon(q)}$$

are bijective. In the smooth category for $q$$q$ even $W^{3q}_{q-1,q}$$W^{3q}_{q-1,q}$ is not injective (see the next subsection), and $W^{3q+1}_{q,q}$$W^{3q+1}_{q,q}$ is not surjective [Boechat1971], [Skopenkov2016f], and $W^7_{2,2}$$W^7_{2,2}$ is not injective [Skopenkov2016f].

6.3 Classification in the presence of smoothly knotted spheres

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.

$\displaystyle E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)}$ $$\displaystyle E^{3s}_D(S^{2s-1})\cong\Z_{\varepsilon(s)}$$

for each $s>1$$s>1$ [Haefliger1966]. The following result for $n=2k+3$$n=2k+3$ was obtained using the Boéchat-Haefliger formula for the smoothing obstruction [Boechat&Haefliger1970], [Boechat1971]. Using that formula one can define the higher-dimensional Kreck invariant [Skopenkov2008].

Theorem 6.3 [Skopenkov2008]. 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 each $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\}$.

E.g. by Theorem 6.3 the Whitney invariant $W:E^{6l}_D(S^{2l-1}\times S^{2l})\to\Zz$$W:E^{6l}_D(S^{2l-1}\times S^{2l})\to\Zz$ is surjective and for each $u\in\Zz$$u\in\Zz$ there is a 1-1 correspondence $W^{-1}u\to\Zz_u$$W^{-1}u\to\Zz_u$.

6.4 Classification further below the stable range

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 remarks on $E^{2n-1}(N)$$E^{2n-1}(N)$ in $\S$$\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]. 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

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

For $k=0$$k=0$ this is the same as General Position Theorem 2.1 [Skopenkov2016c] (because $V_{2n+1,n+1}$$V_{2n+1,n+1}$ is $(n-1)$$(n-1)$-connected). For $k=1$$k=1$ this is covered by Theorem 6.2; for $k\ge2$$k\ge2$ it is not.

E.g. by Theorem 6.4 there is a 1-1 correspondence $E^m(S^p\times S^q)\to\pi_p(V_{m,p+q+1})\oplus\pi_q(V_{m,p+q+1})$$E^m(S^p\times S^q)\to\pi_p(V_{m,p+q+1})\oplus\pi_q(V_{m,p+q+1})$ for $m\ge2q+3$$m\ge2q+3$ and $2m\ge3q+3p+4$$2m\ge3q+3p+4$. For a generalization see [Skopenkov2016k] (to knotted tori) and [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 each $M>n$$M>n$.

7 An orientation on the self-intersection set

Let $f:N\to\Rr^m$$f:N\to\Rr^m$ be a general position smooth map of an oriented $n$$n$-manifold $N$$N$. Assume that $m\ge n+2$$m\ge n+2$ so that the closure $Cl\Sigma(f)$$Cl\Sigma(f)$ of the self-intersection set of $f$$f$ has codimension 2 singularities.

Definition 7.1 (a natural orientation on $\Sigma(f)$$\Sigma(f)$). Take points $x,y\in N$$x,y\in N$ away from the singularities of $\Sigma(f)$$\Sigma(f)$ and such that $fx=fy$$fx=fy$. Then a $(2n-m)$$(2n-m)$-base $\xi_x$$\xi_x$ tangent to $\Sigma(f)$$\Sigma(f)$ at $x$$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)$$\Sigma(f)$ at $y$$y$. Since $N$$N$ is orientable, we can take positive $(m-n)$$(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$.

We remark that

• a change of the orientation of $N$$N$ forces changes of the signs of $\eta_x$$\eta_x$ and $\eta_y$$\eta_y$ and so does not change the orientation of $\Sigma(f)$$\Sigma(f)$.

• the natural orientation on $\Sigma(f)$$\Sigma(f)$ need not extend to $Cl\Sigma(f)$$Cl\Sigma(f)$: take the cone $D^3\to\Rr^5$$D^3\to\Rr^5$ over a general position map $S^2\to\Rr^4$$S^2\to\Rr^4$ having only one self-intersection point.

• the natural orientation on $\Sigma(f)$$\Sigma(f)$ extends to $Cl\Sigma(f)$$Cl\Sigma(f)$ if $m-n$$m-n$ is odd [Hudson1969, Lemma 11.4].

Definition 7.2 (a natural orientation on $f\Sigma(f)$$f\Sigma(f)$ for $m-n$$m-n$ even). Take a $(2n-m)$$(2n-m)$-base $\xi$$\xi$ at a point $x\in f\Sigma(f)$$x\in f\Sigma(f)$ away from the singularities of $f\Sigma(f)$$f\Sigma(f)$. Since $N$$N$ is orientable, we can take a positive $(m-n)$$(m-n)$-base $\eta_+$$\eta_+$ normal to $f\Sigma(f)$$f\Sigma(f)$ in one sheet of $f(N)$$f(N)$. Analogously construct an $(m-n)$$(m-n)$-base $\eta_-$$\eta_-$ for the other sheet of $f(N)$$f(N)$. Since $m-n$$m-n$ is even, the orientation of the base $(\xi,\eta_+,\eta_-)$$(\xi,\eta_+,\eta_-)$ of $\Rr^m$$\Rr^m$ does not depend on choosing the first and the other sheet of $f(N)$$f(N)$. 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)$$f\Sigma(f)$.

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