Embeddings just below the stable range: classification

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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 theory of embeddings.

Recall the unknotting theorem that if $N$${{Stub}} == Introduction == <wikitex>; This page is intended not only for specialists in embeddings, but also for mathematician from other areas who want to apply or to learn theory of embeddings. Recall the [[High codimension embeddings#Unknotting theorems|unknotting theorem]] that if $N$ is a connected manifold of dimension $n>1$, and $m \ge2n+1$, then every two embeddings $N \to\Rr^m$ are isotopic. In this page we summarise the situation for $m=2n\ge6$ and some more general situations. See [[High_codimension_embeddings|general introduction on embeddings, notation and conventions]]. </wikitex> == Classification == <wikitex>; {{beginthm|Theorem}}\label{th4} Let $N$ be a closed connected $n$-manifold. The [[Embeddings just below the stable range: classification#The Whitney invariant|Whitney invariant]] $$W:E^{2n}(N)\to\left\{\begin{array}{cc} H_1(N;\Zz_{(n-1)})& \mbox{either $n$ is odd or $N$ is orientable}\ \Zz\oplus\Zz_2^{s-1}&\mbox{$n$ is even, $N$ is non-orientable and } H_1(N,\Zz_2)\cong\Zz_2^s\end{array}\right.$$ is bijective if either $n\ge4$ or $n=3$ and CAT=PL. {{endthm}} This is proved in \cite{Haefliger1962b}, \cite{Haefliger&Hirsch1963}, \cite{Weber1967}, \cite{Bausum1975}, \cite{Vrabec1977} (a minor miscalculation for the non-orientable case being corrected only in \cite{Vrabec1977}). The classification of ''smooth'' [[Embeddings of 3-manifolds in 6-space|embeddings of 3-manifolds in $\Rr^6$]] is more complicated. For embeddings of $n$-manifolds in $\Rr^{2n-1}$ see [[Embeddings_of_4-manifolds_in_7-space|the case of 4-manifolds]], \cite{Yasui1984}, for $n\ge5$ and \cite{Saeki1999}, \cite{Skopenkov2010}, \cite{Tonkonog2010} for non-closed manifolds. Theorem \ref{th4} is generalized to [[Embeddings just below the stable range: classification#A generalization to highly-connected manifolds|a description]] of $E^{2n-k}(N)$ for closed $k$-connected $n$-manifolds $N$. </wikitex> == Examples == <wikitex>; Together with [[Knots,_i.e._embeddings_of_spheres#The_Haefliger_trefoil_knot|the Haefliger knotted sphere]], examples of Hudson tori presented below were the first examples of embeddings in codimension greater than 2 not isotopic to the standard embedding. (Hudson's construction \cite{Hudson1963} was not as explicit as those below.) For $m\ge n+2$ we define the ''standard embedding'' $S^p\times S^{n-p}\to\Rr^m$ as the composition $S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}=\Rr^{n+2}\to\Rr^m$ of standard embeddings. Let $*:=(-1,0,\ldots,0)\in S^{n-1}$. </wikitex> === Hudson tori=== <wikitex>; In this subsection we construct, for $a\in\Zz$ and $n\ge2$, an embedding $$\Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.$$ The reader might first consider the case $n=2$. {{beginthm|Definition}}\label{dh11} (This construction, as opposed to Definition \ref{dh1n}, works for $n=1$.) Take the standard embeddings D^{n+1}\times S^{n-1}\subset\Rr^{2n}$ (where $ means homothety with coefficient 2) and $\partial D^2\subset\partial D^{n+1}$. Take embedded sphere and embedded torus $\partial D^{n+1}\times *\subset 2D^{n+1}\times S^{n-1}\subset\Rr^{2n}\quad\text{and}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$ 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 [[High_codimension_embeddings#Embedded connected sum|unlinked embedded connected sum]] this is a ''linked'' embedded connected sum, i.e. connected sum of two embeddings whose images are ''not'' contained in disjoint cubes.) {{endthm}} {{beginthm|Definition}}\label{dh1n} For $a\in\Zz$ instead of \partial D^{n+1}\times*$ we take $|a|$ copies $(1+\frac1k)\partial D^{n+1}\times*$ ($k=1,\dots,|a|$) of $n$-sphere outside $D^{n+1}\times S^{n-1}$ `parallel' to $\partial D^{n+1}\times *$, with 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 $(k+1)$-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 <!--the abbreivation of embedding $g$ from Definition \ref{dh1n}--> $$S^n\overset g\to S^{2n}-D^{n+1}\times S^{n-1}\sim S^{2n}-S^{n-1}\sim S^n \quad\text{which is}\quad a\in\pi_n(S^n)\cong\Zz.$$ 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 $a\equiv a'\mod2$. {{endthm}} Proposition \ref{pr3} follows by calculation of [[Embeddings_just_below_the_stable_range:_classification#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 $a\not\equiv a'\mod2$. 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{Vrabec1977}, \S6). {{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 <script type="math/tex">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$$N \to\Rr^m$ are isotopic. In this page we summarise the situation for $m=2n\ge6$$m=2n\ge6$ and some more general situations.

See general introduction on embeddings, notation and conventions.

2 Classification

Theorem 2.1. Let $N$$N$ be a closed connected $n$$n$-manifold. The Whitney invariant

$\displaystyle W:E^{2n}(N)\to\left\{\begin{array}{cc} H_1(N;\Zz_{(n-1)})& \mbox{either $n$ is odd or $N$ is orientable}\\ \Zz\oplus\Zz_2^{s-1}&\mbox{$n$ is even, $N$ is non-orientable and } H_1(N,\Zz_2)\cong\Zz_2^s\end{array}\right.$ $$\displaystyle W:E^{2n}(N)\to\left\{\begin{array}{cc} H_1(N;\Zz_{(n-1)})& \mbox{either $n$ is odd or $N$ is orientable}\\ \Zz\oplus\Zz_2^{s-1}&\mbox{$n$ is even, $N$ is non-orientable and } H_1(N,\Zz_2)\cong\Zz_2^s\end{array}\right.$$

is bijective if either $n\ge4$$n\ge4$ or $n=3$$n=3$ and CAT=PL.

This is proved in [Haefliger1962b], [Haefliger&Hirsch1963], [Weber1967], [Bausum1975], [Vrabec1977] (a minor miscalculation for the non-orientable case being corrected only in [Vrabec1977]).

The classification of smooth embeddings of 3-manifolds in $\Rr^6$$\Rr^6$ is more complicated.

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

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

3 Examples

Together with the Haefliger knotted sphere, examples of Hudson tori presented below were the first examples of embeddings in codimension greater than 2 not isotopic to the standard embedding. (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 $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$ of standard embeddings.

Let $*:=(-1,0,\ldots,0)\in S^{n-1}$$*:=(-1,0,\ldots,0)\in S^{n-1}$.

3.1 Hudson tori

In this subsection we construct, for $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. (This construction, as opposed to Definition 3.2, works for $n=1$$n=1$.) Take the standard embeddings $2D^{n+1}\times S^{n-1}\subset\Rr^{2n}$$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 embedded sphere and embedded torus

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

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 this is a linked embedded connected sum, i.e. connected sum of two embeddings whose images are not contained in disjoint cubes.)

Definition 3.2. For $a\in\Zz$$a\in\Zz$ instead of $2\partial D^{n+1}\times*$$2\partial D^{n+1}\times*$ we take $|a|$$|a|$ copies $(1+\frac1k)\partial D^{n+1}\times*$$(1+\frac1k)\partial D^{n+1}\times*$ ($k=1,\dots,|a|$$k=1,\dots,|a|$) of $n$$n$-sphere outside $D^{n+1}\times S^{n-1}$$D^{n+1}\times S^{n-1}$ `parallel' to $\partial D^{n+1}\times *$$\partial D^{n+1}\times *$, with 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$$k$-th copy to $(k+1)$$(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')$$\Hud_n(a')$ if and only if $a\equiv a'\mod2$$a\equiv a'\mod2$.

Proposition 3.3 follows by calculation of the Whitney invariant (Remark 4.3.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')$$\Hud_2(a')$ if $a\not\equiv a'\mod2$$a\not\equiv a'\mod2$. 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], \S6).

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 Figure 2.2 of [Skopenkov2006]. 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 by two points lying in $D^n\times x$$D^n\times x$, i.e., by the image of an embedding $S^0\times x\to D^n\times x$$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 Figure 2.3 of [Skopenkov2006].) 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 analogous proof.

(b) 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$$n=2$, or are smoothly isotopic for $n=3$$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 4.3.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$$p=0$ this corresponds to the Zeeman construction and its composition with the second unframed Kirby move. It would be interesting to know if links $\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$$\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 could be further generalized.

3.2 Action by linked embedded connected sum

In this subsection we generalize the construction of 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}$$f_0:N\to\Rr^{2n}$ and $a\in H_1(N)$$a\in H_1(N)$, 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$$n$-sphere representing homology Alexander dual of $a$$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 of $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 certain arc joining their images).

Take a vector field on $S^1\times*$$S^1\times*$ normal to $S^1\times S^{n-1}$$S^1\times S^{n-1}$. Extend $S^1\times*$$S^1\times*$ along this vector field to a map $\overline a:D^2\to\Rr^{2n}$$\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$$\pi_1(V_{2n-2,n-1})=0$. Hence the standard framing of $S^1\times*$$S^1\times*$ 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 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 4.1 of the Whitney invariant, $W(f_a)$$W(f_a)$ is $a$$a$ or $a\mod2$$a\mod2$. Thus

• 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;
• linked embedded connected sum defines a free transitive action of $H_1(N;\Zz_{(n-1)})$$H_1(N;\Zz_{(n-1)})$ on $E^{2n}(N)$$E^{2n}(N)$, unless $n=3$$n=3$ and CAT=DIFF.

Parametric connected sum also defines a free transitive action of $H_1(N;\Zz_{(n-1)})$$H_1(N;\Zz_{(n-1)})$ on $E^{2n}(N)$$E^{2n}(N)$ for $n\ge4$$n\ge4$ [Skopenkov2014], Remark 18.a.

4 The Whitney invariant

Let $N$$N$ be a closed $n$$n$-manifold and $f,f_0:N\to\Rr^m$$f,f_0:N\to\Rr^m$ embeddings. Roughly speaking, $W(f,f_0)=W_{f_0}(f)=W(f)$$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 4.2 in the smooth category, following [Skopenkov2010]. The definition in the PL category is analogous [Hudson1969], \S12, [Vrabec1977], p. 145, [Skopenkov2006], \S2.4. Before we present a simpler Definition 4.1 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 4.1. 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$$N_0$ has an $(m-n-1)$$(m-n-1)$-dimensional spine. Therefor 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

Tex syntax error

$\Zz_2$ 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:

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

The orientation on $f\cap F$$f\cap F$ (which extends to $Cl(f\cap F)$$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 this 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$$\Rr^m$, it follows that this condition indeed defines an orientation on $f\cap F$$f\cap F$.

Definition 4.2. 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

Tex syntax error

$\Zz_2$ 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$$m-n$ odd it has a natural orientation and so carries a homology class with $\Zz$$\Zz$ coefficients. Define the Whitney invariant to be the homology class:

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

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 4.3. (a) The Whitney invariant is well-defined by Definition 4.2, i.e. independent of the choice of $H$$H$, analogously to [Skopenkov2006], \S2.4. (The orientation is defined for each $m,n$$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

Tex syntax error

$\Zz_2$-coefficients)

(b) Definition 4.1 is equivalent to Definition 4.2. (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 4.1, 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) Clearly, $W(f)$$W(f)$ is not changed throughout isotopy of $f$$f$. Hence it gives a map $W:E^m(N)\to H_{2n-m+1}(N;\Zz_{(m-n-1)})$$W:E^m(N)\to H_{2n-m+1}(N;\Zz_{(m-n-1)})$.

(d) 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$).

(e) 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$.

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

(e) For $m=2n+1$$m=2n+1$ the Whitney invariant is the collection of pairwise linking coefficients of the components of $N$$N$, cf. Definition 3.2 for $m=2p+1$$m=2p+1$ and $p=q$$p=q$.

5 A generalization to highly-connected manifolds

Let $N$$N$ be a closed $k$$k$-connected $n$$n$-manifold. Recall the unknotting theorem that every two embeddings $N \to\Rr^m$$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 present description of $E^{2n-k}(N)$$E^{2n-k}(N)$ generalizing Theorem 2.1, and its generalization to $E^m(N)$$E^m(N)$ for $m\ge2n-2k+1$$m\ge2n-2k+1$.

Examples are Hudson tori $\Hud_{m,n,p}(a), \Hud'_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$$\Hud_{m,n,p}(a), \Hud'_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$. An action of $H_{k+1}(N;\Z_{(n-k-1)})$$H_{k+1}(N;\Z_{(n-k-1)})$ on $E^{2n-k}(N)$$E^{2n-k}(N)$ for $n\ge2k+4$$n\ge2k+4$ is defined either by analogue of linked embedded connected sum, cf. [Skopenkov2010], Definition 1.4, or by parametric connected sum [Skopenkov2014], Remark 18.a. Both these actions are free transitive by Theorem 5.1 below. If $n=2k+3$$n=2k+3$, then analogue of linked embedded connected sum gives only a construction of embeddings $f_a:N\to\Rr^{2n-k}$$f_a:N\to\Rr^{2n-k}$ for each $a\in H_{k+1}(N;\Z)$$a\in H_{k+1}(N;\Z)$ but not a well-defined action of $H_{k+1}(N;\Z_{(n-k-1)})$$H_{k+1}(N;\Z_{(n-k-1)})$ on $E^{2n-k}(N)$$E^{2n-k}(N)$.

5.1 Classification just below the stable range

Theorem 5.1. Let $N$$N$ be a closed orientable homologically $k$$k$-connected $n$$n$-manifold, $k\ge0$$k\ge0$. Then the Whitney invariant

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

is a bijection, provided $n\ge k+3$$n\ge k+3$ or $n\ge2k+4$$n\ge2k+4$ in the PL or DIFF categories, respectively.

Theorem 5.1 was proved in [Haefliger&Hirsch1963], [Hudson1969], \S11, [Boechat&Haefliger1970], [Boechat1971], [Vrabec1977] homotopically $k$$k$-connected manifolds. The proof works for homologically $k$$k$-connected manifolds.

For $k=0$$k=0$ this is covered by Theorem 2.1; for $k\ge1$$k\ge1$ it is not. The PL case of Theorem 5.1 gives nothing but the Unknotting Spheres Theorem for $k+3\le n\le2k+1$$k+3\le n\le2k+1$.

By Theorem 5.1 the Whitney invariant $W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{(q)}$$W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{(q)}$ is bijective for $1\le p\le q-2$$1\le p\le q-2$. It is in fact a group isomorphism (for the group structure introduced in [Skopenkov2006a], [Skopenkov2015], [Skopenkov2015]). The generator is Hudson torus $\Hud(1)$$\Hud(1)$. Also, for $q\ge2$$q\ge2$ by Theorem 5.1 the Whitney invariants

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

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

5.2 Classification in the presence of smoothly knotted spheres

Because of the existence of knots the analogues of Theorem 5.1 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)$$E^{2n-k}(N)$ is much harder: for 40 years the only known complete readily calculable classification results were for homology sphere $N$$N$. The following result for $n=2k+3$$n=2k+3$ was obtained using the Bo\'echat-Haefliger formula for the smoothing obstruction [Boechat1971].

Theorem 5.2. [Skopenkov2008] Let $N$$N$ be a closed homologically $(2l-2)$$(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 analogue of 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 5.2 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$.

5.3 Classification further below the stable range

How to describe $E^m(N)$$E^m(N)$ for $m<2n-k$$m<2n-k$? See remarks on $E^{2n-1}(N)$$E^{2n-1}(N)$. Some estimations of $E^{2n-k-1}(N)$$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 5.3. [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$$k=0$ this is the same as General Position Theorem 2.1 (because $V_{2n+1,n+1}$$V_{2n+1,n+1}$ is $(n-1)$$(n-1)$-connected). For $k=1$$k=1$ this is covered by Theorem 5.1; for $k\ge2$$k\ge2$ it is not.

E.g. by Theorem 5.3 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 Knotted tori and [Skopenkov2002].

Observe that in Theorem 5.3 $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$.

6 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. Then

• (1) $\Sigma(f)$$\Sigma(f)$ has a natural orientation.
• (2) the natural orientation on $\Sigma(f)$$\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$$S^2\to\Rr^4$ having only one self-intersection point.
• (3) the natural orientation on $\Sigma(f)$$\Sigma(f)$ extend to $Cl\Sigma(f)$$Cl\Sigma(f)$ if $m-n$$m-n$ is odd [Hudson1969], Lemma 11.4.
• (4) $f\Sigma(f)$$f\Sigma(f)$ has a natural orientation if $m-n$$m-n$ is even.

Let us prove (1). Take points $x,y\in N$$x,y\in N$ outside singularities of $\Sigma(f)$$\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$$\eta_y$. (Note that a change of the orientation of $N$$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)$.)

Let us prove (4). Take a $(2n-m)$$(2n-m)$-base $\xi$$\xi$ at a point $x\in f\Sigma(f)$$x\in f\Sigma(f)$ outside singularities of $f\Sigma(f)$$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_-)$$(\xi,\eta_+,\eta_-)$. (Note that a change of the orientation of $N$$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)$.)

7 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$$N \to\Rr^m$ are isotopic. In this page we summarise the situation for $m=2n\ge6$$m=2n\ge6$ and some more general situations.

See general introduction on embeddings, notation and conventions.

2 Classification

Theorem 2.1. Let $N$$N$ be a closed connected $n$$n$-manifold. The Whitney invariant

$\displaystyle W:E^{2n}(N)\to\left\{\begin{array}{cc} H_1(N;\Zz_{(n-1)})& \mbox{either $n$ is odd or $N$ is orientable}\\ \Zz\oplus\Zz_2^{s-1}&\mbox{$n$ is even, $N$ is non-orientable and } H_1(N,\Zz_2)\cong\Zz_2^s\end{array}\right.$ $$\displaystyle W:E^{2n}(N)\to\left\{\begin{array}{cc} H_1(N;\Zz_{(n-1)})& \mbox{either $n$ is odd or $N$ is orientable}\\ \Zz\oplus\Zz_2^{s-1}&\mbox{$n$ is even, $N$ is non-orientable and } H_1(N,\Zz_2)\cong\Zz_2^s\end{array}\right.$$

is bijective if either $n\ge4$$n\ge4$ or $n=3$$n=3$ and CAT=PL.

This is proved in [Haefliger1962b], [Haefliger&Hirsch1963], [Weber1967], [Bausum1975], [Vrabec1977] (a minor miscalculation for the non-orientable case being corrected only in [Vrabec1977]).

The classification of smooth embeddings of 3-manifolds in $\Rr^6$$\Rr^6$ is more complicated.

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

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

3 Examples

Together with the Haefliger knotted sphere, examples of Hudson tori presented below were the first examples of embeddings in codimension greater than 2 not isotopic to the standard embedding. (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 $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$ of standard embeddings.

Let $*:=(-1,0,\ldots,0)\in S^{n-1}$$*:=(-1,0,\ldots,0)\in S^{n-1}$.

3.1 Hudson tori

In this subsection we construct, for $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. (This construction, as opposed to Definition 3.2, works for $n=1$$n=1$.) Take the standard embeddings $2D^{n+1}\times S^{n-1}\subset\Rr^{2n}$$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 embedded sphere and embedded torus

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

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 this is a linked embedded connected sum, i.e. connected sum of two embeddings whose images are not contained in disjoint cubes.)

Definition 3.2. For $a\in\Zz$$a\in\Zz$ instead of $2\partial D^{n+1}\times*$$2\partial D^{n+1}\times*$ we take $|a|$$|a|$ copies $(1+\frac1k)\partial D^{n+1}\times*$$(1+\frac1k)\partial D^{n+1}\times*$ ($k=1,\dots,|a|$$k=1,\dots,|a|$) of $n$$n$-sphere outside $D^{n+1}\times S^{n-1}$$D^{n+1}\times S^{n-1}$ `parallel' to $\partial D^{n+1}\times *$$\partial D^{n+1}\times *$, with 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$$k$-th copy to $(k+1)$$(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')$$\Hud_n(a')$ if and only if $a\equiv a'\mod2$$a\equiv a'\mod2$.

Proposition 3.3 follows by calculation of the Whitney invariant (Remark 4.3.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')$$\Hud_2(a')$ if $a\not\equiv a'\mod2$$a\not\equiv a'\mod2$. 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], \S6).

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 Figure 2.2 of [Skopenkov2006]. 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 by two points lying in $D^n\times x$$D^n\times x$, i.e., by the image of an embedding $S^0\times x\to D^n\times x$$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 Figure 2.3 of [Skopenkov2006].) 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 analogous proof.

(b) 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$$n=2$, or are smoothly isotopic for $n=3$$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 4.3.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$$p=0$ this corresponds to the Zeeman construction and its composition with the second unframed Kirby move. It would be interesting to know if links $\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$$\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 could be further generalized.

3.2 Action by linked embedded connected sum

In this subsection we generalize the construction of 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}$$f_0:N\to\Rr^{2n}$ and $a\in H_1(N)$$a\in H_1(N)$, 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$$n$-sphere representing homology Alexander dual of $a$$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 of $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 certain arc joining their images).

Take a vector field on $S^1\times*$$S^1\times*$ normal to $S^1\times S^{n-1}$$S^1\times S^{n-1}$. Extend $S^1\times*$$S^1\times*$ along this vector field to a map $\overline a:D^2\to\Rr^{2n}$$\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$$\pi_1(V_{2n-2,n-1})=0$. Hence the standard framing of $S^1\times*$$S^1\times*$ 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 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 4.1 of the Whitney invariant, $W(f_a)$$W(f_a)$ is $a$$a$ or $a\mod2$$a\mod2$. Thus

• 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;
• linked embedded connected sum defines a free transitive action of $H_1(N;\Zz_{(n-1)})$$H_1(N;\Zz_{(n-1)})$ on $E^{2n}(N)$$E^{2n}(N)$, unless $n=3$$n=3$ and CAT=DIFF.

Parametric connected sum also defines a free transitive action of $H_1(N;\Zz_{(n-1)})$$H_1(N;\Zz_{(n-1)})$ on $E^{2n}(N)$$E^{2n}(N)$ for $n\ge4$$n\ge4$ [Skopenkov2014], Remark 18.a.

4 The Whitney invariant

Let $N$$N$ be a closed $n$$n$-manifold and $f,f_0:N\to\Rr^m$$f,f_0:N\to\Rr^m$ embeddings. Roughly speaking, $W(f,f_0)=W_{f_0}(f)=W(f)$$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 4.2 in the smooth category, following [Skopenkov2010]. The definition in the PL category is analogous [Hudson1969], \S12, [Vrabec1977], p. 145, [Skopenkov2006], \S2.4. Before we present a simpler Definition 4.1 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 4.1. 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$$N_0$ has an $(m-n-1)$$(m-n-1)$-dimensional spine. Therefor 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

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$\Zz_2$ 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:

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

The orientation on $f\cap F$$f\cap F$ (which extends to $Cl(f\cap F)$$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 this 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$$\Rr^m$, it follows that this condition indeed defines an orientation on $f\cap F$$f\cap F$.

Definition 4.2. 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

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$\Zz_2$ 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$$m-n$ odd it has a natural orientation and so carries a homology class with $\Zz$$\Zz$ coefficients. Define the Whitney invariant to be the homology class:

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

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 4.3. (a) The Whitney invariant is well-defined by Definition 4.2, i.e. independent of the choice of $H$$H$, analogously to [Skopenkov2006], \S2.4. (The orientation is defined for each $m,n$$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

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$\Zz_2$-coefficients)

(b) Definition 4.1 is equivalent to Definition 4.2. (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 4.1, 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) Clearly, $W(f)$$W(f)$ is not changed throughout isotopy of $f$$f$. Hence it gives a map $W:E^m(N)\to H_{2n-m+1}(N;\Zz_{(m-n-1)})$$W:E^m(N)\to H_{2n-m+1}(N;\Zz_{(m-n-1)})$.

(d) 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$).

(e) 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$.

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

(e) For $m=2n+1$$m=2n+1$ the Whitney invariant is the collection of pairwise linking coefficients of the components of $N$$N$, cf. Definition 3.2 for $m=2p+1$$m=2p+1$ and $p=q$$p=q$.

5 A generalization to highly-connected manifolds

Let $N$$N$ be a closed $k$$k$-connected $n$$n$-manifold. Recall the unknotting theorem that every two embeddings $N \to\Rr^m$$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 present description of $E^{2n-k}(N)$$E^{2n-k}(N)$ generalizing Theorem 2.1, and its generalization to $E^m(N)$$E^m(N)$ for $m\ge2n-2k+1$$m\ge2n-2k+1$.

Examples are Hudson tori $\Hud_{m,n,p}(a), \Hud'_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$$\Hud_{m,n,p}(a), \Hud'_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$. An action of $H_{k+1}(N;\Z_{(n-k-1)})$$H_{k+1}(N;\Z_{(n-k-1)})$ on $E^{2n-k}(N)$$E^{2n-k}(N)$ for $n\ge2k+4$$n\ge2k+4$ is defined either by analogue of linked embedded connected sum, cf. [Skopenkov2010], Definition 1.4, or by parametric connected sum [Skopenkov2014], Remark 18.a. Both these actions are free transitive by Theorem 5.1 below. If $n=2k+3$$n=2k+3$, then analogue of linked embedded connected sum gives only a construction of embeddings $f_a:N\to\Rr^{2n-k}$$f_a:N\to\Rr^{2n-k}$ for each $a\in H_{k+1}(N;\Z)$$a\in H_{k+1}(N;\Z)$ but not a well-defined action of $H_{k+1}(N;\Z_{(n-k-1)})$$H_{k+1}(N;\Z_{(n-k-1)})$ on $E^{2n-k}(N)$$E^{2n-k}(N)$.

5.1 Classification just below the stable range

Theorem 5.1. Let $N$$N$ be a closed orientable homologically $k$$k$-connected $n$$n$-manifold, $k\ge0$$k\ge0$. Then the Whitney invariant

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

is a bijection, provided $n\ge k+3$$n\ge k+3$ or $n\ge2k+4$$n\ge2k+4$ in the PL or DIFF categories, respectively.

Theorem 5.1 was proved in [Haefliger&Hirsch1963], [Hudson1969], \S11, [Boechat&Haefliger1970], [Boechat1971], [Vrabec1977] homotopically $k$$k$-connected manifolds. The proof works for homologically $k$$k$-connected manifolds.

For $k=0$$k=0$ this is covered by Theorem 2.1; for $k\ge1$$k\ge1$ it is not. The PL case of Theorem 5.1 gives nothing but the Unknotting Spheres Theorem for $k+3\le n\le2k+1$$k+3\le n\le2k+1$.

By Theorem 5.1 the Whitney invariant $W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{(q)}$$W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{(q)}$ is bijective for $1\le p\le q-2$$1\le p\le q-2$. It is in fact a group isomorphism (for the group structure introduced in [Skopenkov2006a], [Skopenkov2015], [Skopenkov2015]). The generator is Hudson torus $\Hud(1)$$\Hud(1)$. Also, for $q\ge2$$q\ge2$ by Theorem 5.1 the Whitney invariants

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

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

5.2 Classification in the presence of smoothly knotted spheres

Because of the existence of knots the analogues of Theorem 5.1 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)$$E^{2n-k}(N)$ is much harder: for 40 years the only known complete readily calculable classification results were for homology sphere $N$$N$. The following result for $n=2k+3$$n=2k+3$ was obtained using the Bo\'echat-Haefliger formula for the smoothing obstruction [Boechat1971].

Theorem 5.2. [Skopenkov2008] Let $N$$N$ be a closed homologically $(2l-2)$$(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 analogue of 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 5.2 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$.

5.3 Classification further below the stable range

How to describe $E^m(N)$$E^m(N)$ for $m<2n-k$$m<2n-k$? See remarks on $E^{2n-1}(N)$$E^{2n-1}(N)$. Some estimations of $E^{2n-k-1}(N)$$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 5.3. [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$$k=0$ this is the same as General Position Theorem 2.1 (because $V_{2n+1,n+1}$$V_{2n+1,n+1}$ is $(n-1)$$(n-1)$-connected). For $k=1$$k=1$ this is covered by Theorem 5.1; for $k\ge2$$k\ge2$ it is not.

E.g. by Theorem 5.3 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 Knotted tori and [Skopenkov2002].

Observe that in Theorem 5.3 $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$.

6 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. Then

• (1) $\Sigma(f)$$\Sigma(f)$ has a natural orientation.
• (2) the natural orientation on $\Sigma(f)$$\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$$S^2\to\Rr^4$ having only one self-intersection point.
• (3) the natural orientation on $\Sigma(f)$$\Sigma(f)$ extend to $Cl\Sigma(f)$$Cl\Sigma(f)$ if $m-n$$m-n$ is odd [Hudson1969], Lemma 11.4.
• (4) $f\Sigma(f)$$f\Sigma(f)$ has a natural orientation if $m-n$$m-n$ is even.

Let us prove (1). Take points $x,y\in N$$x,y\in N$ outside singularities of $\Sigma(f)$$\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$$\eta_y$. (Note that a change of the orientation of $N$$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)$.)

Let us prove (4). Take a $(2n-m)$$(2n-m)$-base $\xi$$\xi$ at a point $x\in f\Sigma(f)$$x\in f\Sigma(f)$ outside singularities of $f\Sigma(f)$$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_-)$$(\xi,\eta_+,\eta_-)$. (Note that a change of the orientation of $N$$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 Figure 2.2 of \cite{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 by two points lying in $D^n\times x$, i.e., by 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 Figure 2.3 of \cite{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 analogous proof. (b) 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 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 [[Links,_i.e._embeddings_of_non-connected_manifolds#The_Zeeman_construction_and_linking_coefficient|the Zeeman construction]] and its composition with ''the second unframed Kirby move''. It would be interesting to know if links $\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$ are isotopic, cf. \cite{Skopenkov2015a}, Remark 2.9.b. These constructions could be [[Knotted_tori#Examples|further generalized]]. {{endthm}} ===Action by linked embedded connected sum=== ; In this subsection we generalize the construction of Hudson torus $\Hud(a)$. For $n\ge3$, a closed connected orientable $n$-manifold $N$, an embedding $f_0:N\to\Rr^{2n}$ and $a\in H_1(N)$, 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 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 of $S^1\times*\subset S^1\times S^{n-1}$ 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 certain arc joining their images). Take a vector field on $S^1\times*$ normal to $S^1\times S^{n-1}$. Extend $S^1\times*$ 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 $S^1\times*$ 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 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 [[Embeddings_just_below_the_stable_range:_classification#The_Whitney_invariant|the Whitney invariant]], $W(f_a)$ is $a$ or $a\mod2$. Thus * all isotopy classes of embeddings $N\to\Rr^{2n}$ can be obtained from any chosen embedding $f_0$ by the above construction; * linked embedded connected sum defines a free transitive action of $H_1(N;\Zz_{(n-1)})$ on $E^{2n}(N)$, unless $n=3$ and CAT=DIFF. [[Parametric_connected_sum#Applications|Parametric connected sum]] also defines a free transitive action of $H_1(N;\Zz_{(n-1)})$ on $E^{2n}(N)$ for $n\ge4$ \cite{Skopenkov2014}, Remark 18.a. == The Whitney invariant == ; Let $N$ be a closed $n$-manifold and $f,f_0:N\to\Rr^m$ embeddings. 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{Hudson1969}, \S12, \cite{Vrabec1977}, p. 145, \cite{Skopenkov2006}, \S2.4. Before we present a simpler 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, $N_0$ has an $(m-n-1)$-dimensional spine. Therefor these restrictions are isotopic, cf. \cite{Haefliger&Hirsch1963}, 3.1.b, \cite{Takase2006}, Lemma 2.2. 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: $$ W(f):=[Cl(f\cap F)]\in H_{2n-m+1}(N_0,\partial N_0;\Zz_{(m-n-1)})\cong H_{2n-m+1}(N;\Zz_{(m-n-1)}). $$ The orientation on $f\cap F$ (which extends 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 this 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 $\Rr^m$, it follows that 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 [[Embeddings just below the stable range: classification#An orientation on the self-intersection set|natural orientation]] and so carries a homology class with $\Zz$ coefficients. Define the Whitney invariant to be the homology class: $$ W(f):=[Cl\Sigma(H)]\in H_{2n-m+1}(N\times I;\Zz_{(m-n-1)})\cong H_{2n-m+1}(N;\Zz_{(m-n-1)}). $$ {{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. independent of the choice of $H$, analogously to \cite{Skopenkov2006}, \S2.4. (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) (b) Definition \ref{dw1} is equivalent to 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) Clearly, $W(f)$ is not changed throughout isotopy of $f$. Hence it gives a map $W:E^m(N)\to H_{2n-m+1}(N;\Zz_{(m-n-1)})$. (d) 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 $f\cap F$). (e) For the [[Embeddings just below the stable range: classification#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 $n=2$. (f) $W(f\#g)=W(f)$ for each embeddings $f:N\to\Rr^m$ and $g:S^n\to\Rr^m$. (e) For $m=2n+1$ the Whitney invariant is the collection of pairwise linking coefficients of the components of $N$, cf. [[High_codimension_links#Definitions_of_the_Zeeman_map_and_linking_coefficient|Definition 3.2]] for $m=2p+1$ and $p=q$. {{endthm}} == A generalization to highly-connected manifolds == ; Let $N$ be a closed $k$-connected $n$-manifold. Recall the [[High codimension embeddings#Unknotting theorems|unknotting theorem]] that every two embeddings $N \to\Rr^m$ are isotopic when $m\ge 2n-k+1$ and $n\ge2k+2$. In this section we present description of $E^{2n-k}(N)$ generalizing Theorem \ref{th4}, and its generalization to $E^m(N)$ for $m\ge2n-2k+1$. Examples are [[Embeddings_just_below_the_stable_range:_classification#Hudson_tori|Hudson tori]] $\Hud_{m,n,p}(a), \Hud'_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$. An action of $H_{k+1}(N;\Z_{(n-k-1)})$ on $E^{2n-k}(N)$ for $n\ge2k+4$ is defined either by analogue of [[Embeddings_just_below_the_stable_range:_classification#Action_by_linked_embedded_connected_sum|linked embedded connected sum]], cf. \cite{Skopenkov2010}, Definition 1.4, or by [[Parametric_connected_sum#Applications|parametric connected sum]] \cite{Skopenkov2014}, Remark 18.a. Both these actions are free transitive by Theorem \ref{hico} below. If $n=2k+3$, then analogue of [[Embeddings_just_below_the_stable_range:_classification#Action_by_linked_embedded_connected_sum|linked embedded connected sum]] gives only a construction of embeddings $f_a:N\to\Rr^{2n-k}$ for each $a\in H_{k+1}(N;\Z)$ but not a well-defined action of $H_{k+1}(N;\Z_{(n-k-1)})$ on $E^{2n-k}(N)$. === Classification just below the stable range === ; {{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]] $$W:E^{2n-k}(N)\to H_{k+1}(N,\Zz_{(n-k-1)})$$ is a bijection, provided $n\ge k+3$ or $n\ge2k+4$ in the PL or DIFF categories, respectively. {{endthm}} Theorem \ref{hico} was proved in \cite{Haefliger&Hirsch1963}, \cite{Hudson1969}, \S11, \cite{Boechat&Haefliger1970}, \cite{Boechat1971}, \cite{Vrabec1977} ''homotopically'' $k$-connected manifolds. The proof works for ''homologically'' $k$-connected manifolds. For $k=0$ this is covered by Theorem \ref{th4}; for $k\ge1$ it is not. The PL case of Theorem \ref{hico} gives nothing but [[High codimension embeddings: classification#Unknotting theorems|the Unknotting Spheres Theorem]] for $k+3\le n\le2k+1$. By Theorem \ref{hico} the Whitney invariant $W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{(q)}$ 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$$N \to\Rr^m$ are isotopic. In this page we summarise the situation for $m=2n\ge6$$m=2n\ge6$ and some more general situations.

See general introduction on embeddings, notation and conventions.

2 Classification

Theorem 2.1. Let $N$$N$ be a closed connected $n$$n$-manifold. The Whitney invariant

$\displaystyle W:E^{2n}(N)\to\left\{\begin{array}{cc} H_1(N;\Zz_{(n-1)})& \mbox{either $n$ is odd or $N$ is orientable}\\ \Zz\oplus\Zz_2^{s-1}&\mbox{$n$ is even, $N$ is non-orientable and } H_1(N,\Zz_2)\cong\Zz_2^s\end{array}\right.$ $$\displaystyle W:E^{2n}(N)\to\left\{\begin{array}{cc} H_1(N;\Zz_{(n-1)})& \mbox{either $n$ is odd or $N$ is orientable}\\ \Zz\oplus\Zz_2^{s-1}&\mbox{$n$ is even, $N$ is non-orientable and } H_1(N,\Zz_2)\cong\Zz_2^s\end{array}\right.$$

is bijective if either $n\ge4$$n\ge4$ or $n=3$$n=3$ and CAT=PL.

This is proved in [Haefliger1962b], [Haefliger&Hirsch1963], [Weber1967], [Bausum1975], [Vrabec1977] (a minor miscalculation for the non-orientable case being corrected only in [Vrabec1977]).

The classification of smooth embeddings of 3-manifolds in $\Rr^6$$\Rr^6$ is more complicated.

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

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

3 Examples

Together with the Haefliger knotted sphere, examples of Hudson tori presented below were the first examples of embeddings in codimension greater than 2 not isotopic to the standard embedding. (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 $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$ of standard embeddings.

Let $*:=(-1,0,\ldots,0)\in S^{n-1}$$*:=(-1,0,\ldots,0)\in S^{n-1}$.

3.1 Hudson tori

In this subsection we construct, for $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. (This construction, as opposed to Definition 3.2, works for $n=1$$n=1$.) Take the standard embeddings $2D^{n+1}\times S^{n-1}\subset\Rr^{2n}$$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 embedded sphere and embedded torus

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

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 this is a linked embedded connected sum, i.e. connected sum of two embeddings whose images are not contained in disjoint cubes.)

Definition 3.2. For $a\in\Zz$$a\in\Zz$ instead of $2\partial D^{n+1}\times*$$2\partial D^{n+1}\times*$ we take $|a|$$|a|$ copies $(1+\frac1k)\partial D^{n+1}\times*$$(1+\frac1k)\partial D^{n+1}\times*$ ($k=1,\dots,|a|$$k=1,\dots,|a|$) of $n$$n$-sphere outside $D^{n+1}\times S^{n-1}$$D^{n+1}\times S^{n-1}$ `parallel' to $\partial D^{n+1}\times *$$\partial D^{n+1}\times *$, with 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$$k$-th copy to $(k+1)$$(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')$$\Hud_n(a')$ if and only if $a\equiv a'\mod2$$a\equiv a'\mod2$.

Proposition 3.3 follows by calculation of the Whitney invariant (Remark 4.3.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')$$\Hud_2(a')$ if $a\not\equiv a'\mod2$$a\not\equiv a'\mod2$. 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], \S6).

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 Figure 2.2 of [Skopenkov2006]. 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 by two points lying in $D^n\times x$$D^n\times x$, i.e., by the image of an embedding $S^0\times x\to D^n\times x$$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 Figure 2.3 of [Skopenkov2006].) 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 analogous proof.

(b) 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$$n=2$, or are smoothly isotopic for $n=3$$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 4.3.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$$p=0$ this corresponds to the Zeeman construction and its composition with the second unframed Kirby move. It would be interesting to know if links $\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$$\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 could be further generalized.

3.2 Action by linked embedded connected sum

In this subsection we generalize the construction of 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}$$f_0:N\to\Rr^{2n}$ and $a\in H_1(N)$$a\in H_1(N)$, 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$$n$-sphere representing homology Alexander dual of $a$$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 of $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 certain arc joining their images).

Take a vector field on $S^1\times*$$S^1\times*$ normal to $S^1\times S^{n-1}$$S^1\times S^{n-1}$. Extend $S^1\times*$$S^1\times*$ along this vector field to a map $\overline a:D^2\to\Rr^{2n}$$\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$$\pi_1(V_{2n-2,n-1})=0$. Hence the standard framing of $S^1\times*$$S^1\times*$ 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 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 4.1 of the Whitney invariant, $W(f_a)$$W(f_a)$ is $a$$a$ or $a\mod2$$a\mod2$. Thus

• 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;
• linked embedded connected sum defines a free transitive action of $H_1(N;\Zz_{(n-1)})$$H_1(N;\Zz_{(n-1)})$ on $E^{2n}(N)$$E^{2n}(N)$, unless $n=3$$n=3$ and CAT=DIFF.

Parametric connected sum also defines a free transitive action of $H_1(N;\Zz_{(n-1)})$$H_1(N;\Zz_{(n-1)})$ on $E^{2n}(N)$$E^{2n}(N)$ for $n\ge4$$n\ge4$ [Skopenkov2014], Remark 18.a.

4 The Whitney invariant

Let $N$$N$ be a closed $n$$n$-manifold and $f,f_0:N\to\Rr^m$$f,f_0:N\to\Rr^m$ embeddings. Roughly speaking, $W(f,f_0)=W_{f_0}(f)=W(f)$$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 4.2 in the smooth category, following [Skopenkov2010]. The definition in the PL category is analogous [Hudson1969], \S12, [Vrabec1977], p. 145, [Skopenkov2006], \S2.4. Before we present a simpler Definition 4.1 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 4.1. 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$$N_0$ has an $(m-n-1)$$(m-n-1)$-dimensional spine. Therefor 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

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$\Zz_2$ 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:

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

The orientation on $f\cap F$$f\cap F$ (which extends to $Cl(f\cap F)$$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 this 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$$\Rr^m$, it follows that this condition indeed defines an orientation on $f\cap F$$f\cap F$.

Definition 4.2. 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

Tex syntax error

$\Zz_2$ 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$$m-n$ odd it has a natural orientation and so carries a homology class with $\Zz$$\Zz$ coefficients. Define the Whitney invariant to be the homology class:

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

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 4.3. (a) The Whitney invariant is well-defined by Definition 4.2, i.e. independent of the choice of $H$$H$, analogously to [Skopenkov2006], \S2.4. (The orientation is defined for each $m,n$$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

Tex syntax error

$\Zz_2$-coefficients)

(b) Definition 4.1 is equivalent to Definition 4.2. (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 4.1, 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) Clearly, $W(f)$$W(f)$ is not changed throughout isotopy of $f$$f$. Hence it gives a map $W:E^m(N)\to H_{2n-m+1}(N;\Zz_{(m-n-1)})$$W:E^m(N)\to H_{2n-m+1}(N;\Zz_{(m-n-1)})$.

(d) 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$).

(e) 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$.

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

(e) For $m=2n+1$$m=2n+1$ the Whitney invariant is the collection of pairwise linking coefficients of the components of $N$$N$, cf. Definition 3.2 for $m=2p+1$$m=2p+1$ and $p=q$$p=q$.

5 A generalization to highly-connected manifolds

Let $N$$N$ be a closed $k$$k$-connected $n$$n$-manifold. Recall the unknotting theorem that every two embeddings $N \to\Rr^m$$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 present description of $E^{2n-k}(N)$$E^{2n-k}(N)$ generalizing Theorem 2.1, and its generalization to $E^m(N)$$E^m(N)$ for $m\ge2n-2k+1$$m\ge2n-2k+1$.

Examples are Hudson tori $\Hud_{m,n,p}(a), \Hud'_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$$\Hud_{m,n,p}(a), \Hud'_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$. An action of $H_{k+1}(N;\Z_{(n-k-1)})$$H_{k+1}(N;\Z_{(n-k-1)})$ on $E^{2n-k}(N)$$E^{2n-k}(N)$ for $n\ge2k+4$$n\ge2k+4$ is defined either by analogue of linked embedded connected sum, cf. [Skopenkov2010], Definition 1.4, or by parametric connected sum [Skopenkov2014], Remark 18.a. Both these actions are free transitive by Theorem 5.1 below. If $n=2k+3$$n=2k+3$, then analogue of linked embedded connected sum gives only a construction of embeddings $f_a:N\to\Rr^{2n-k}$$f_a:N\to\Rr^{2n-k}$ for each $a\in H_{k+1}(N;\Z)$$a\in H_{k+1}(N;\Z)$ but not a well-defined action of $H_{k+1}(N;\Z_{(n-k-1)})$$H_{k+1}(N;\Z_{(n-k-1)})$ on $E^{2n-k}(N)$$E^{2n-k}(N)$.

5.1 Classification just below the stable range

Theorem 5.1. Let $N$$N$ be a closed orientable homologically $k$$k$-connected $n$$n$-manifold, $k\ge0$$k\ge0$. Then the Whitney invariant

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

is a bijection, provided $n\ge k+3$$n\ge k+3$ or $n\ge2k+4$$n\ge2k+4$ in the PL or DIFF categories, respectively.

Theorem 5.1 was proved in [Haefliger&Hirsch1963], [Hudson1969], \S11, [Boechat&Haefliger1970], [Boechat1971], [Vrabec1977] homotopically $k$$k$-connected manifolds. The proof works for homologically $k$$k$-connected manifolds.

For $k=0$$k=0$ this is covered by Theorem 2.1; for $k\ge1$$k\ge1$ it is not. The PL case of Theorem 5.1 gives nothing but the Unknotting Spheres Theorem for $k+3\le n\le2k+1$$k+3\le n\le2k+1$.

By Theorem 5.1 the Whitney invariant $W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{(q)}$$W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{(q)}$ is bijective for $1\le p\le q-2$$1\le p\le q-2$. It is in fact a group isomorphism (for the group structure introduced in [Skopenkov2006a], [Skopenkov2015], [Skopenkov2015]). The generator is Hudson torus $\Hud(1)$$\Hud(1)$. Also, for $q\ge2$$q\ge2$ by Theorem 5.1 the Whitney invariants

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

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

5.2 Classification in the presence of smoothly knotted spheres

Because of the existence of knots the analogues of Theorem 5.1 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)$$E^{2n-k}(N)$ is much harder: for 40 years the only known complete readily calculable classification results were for homology sphere $N$$N$. The following result for $n=2k+3$$n=2k+3$ was obtained using the Bo\'echat-Haefliger formula for the smoothing obstruction [Boechat1971].

Theorem 5.2. [Skopenkov2008] Let $N$$N$ be a closed homologically $(2l-2)$$(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 analogue of 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 5.2 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$.

5.3 Classification further below the stable range

How to describe $E^m(N)$$E^m(N)$ for $m<2n-k$$m<2n-k$? See remarks on $E^{2n-1}(N)$$E^{2n-1}(N)$. Some estimations of $E^{2n-k-1}(N)$$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 5.3. [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$$k=0$ this is the same as General Position Theorem 2.1 (because $V_{2n+1,n+1}$$V_{2n+1,n+1}$ is $(n-1)$$(n-1)$-connected). For $k=1$$k=1$ this is covered by Theorem 5.1; for $k\ge2$$k\ge2$ it is not.

E.g. by Theorem 5.3 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 Knotted tori and [Skopenkov2002].

Observe that in Theorem 5.3 $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$.

6 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. Then

• (1) $\Sigma(f)$$\Sigma(f)$ has a natural orientation.
• (2) the natural orientation on $\Sigma(f)$$\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$$S^2\to\Rr^4$ having only one self-intersection point.
• (3) the natural orientation on $\Sigma(f)$$\Sigma(f)$ extend to $Cl\Sigma(f)$$Cl\Sigma(f)$ if $m-n$$m-n$ is odd [Hudson1969], Lemma 11.4.
• (4) $f\Sigma(f)$$f\Sigma(f)$ has a natural orientation if $m-n$$m-n$ is even.

Let us prove (1). Take points $x,y\in N$$x,y\in N$ outside singularities of $\Sigma(f)$$\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$$\eta_y$. (Note that a change of the orientation of $N$$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)$.)

Let us prove (4). Take a $(2n-m)$$(2n-m)$-base $\xi$$\xi$ at a point $x\in f\Sigma(f)$$x\in f\Sigma(f)$ outside singularities of $f\Sigma(f)$$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_-)$$(\xi,\eta_+,\eta_-)$. (Note that a change of the orientation of $N$$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\le q-2$. It is in fact a group isomorphism (for the group structure introduced in \cite{Skopenkov2006a}, \cite{Skopenkov2015}, \cite{Skopenkov2015}). The generator is [[Embeddings just below the stable range: classification#Hudson tori| Hudson torus]] $\Hud(1)$. Also, for $q\ge2$ by Theorem \ref{hico} the Whitney invariants $$W^{3q}_{q-1,q}:E^{3q}_{PL}(S^{q-1}\times S^q)\to\Zz_{(q)} \quad\text{and}\quad W^{3q+1}_{q,q}:E^{3q+1}_{PL}(S^q\times S^q)\to\Zz_{(q)}\oplus \Zz_{(q)}$$ are bijective. In the smooth category for $q$ even $W^{3q}_{q-1,q}$ [[Embeddings_just_below_the_stable_range:_classification#Classification_in_the_presence_of_smoothly_knotted_spheres|is not injective]], and $W^{3q+1}_{q,q}$ is not surjective \cite{Boechat1971}, and $W^7_{2,2}$ [[4-manifolds_in_7-space#Classification|is not injective]]. === Classification in the presence of smoothly knotted spheres === ; Because of the existence of knots 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 sphere $N$. The following result for $n=2k+3$ was obtained using the Bo\'echat-Haefliger formula for the smoothing obstruction \cite{Boechat1971}. {{beginthm|Theorem}}\label{hicod} \cite{Skopenkov2008} Let $N$ be a closed homologically $(2l-2)$-connected $(4l-1)$-manifold. Then [[Embeddings just below the stable range: classification#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 analogue of [[3-manifolds in 6-space#The Kreck invariant|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 to describe $E^m(N)$ for $m<2n-k$? See [[Embeddings_just_below_the_stable_range:_classification#Classification|remarks]] on $E^{2n-1}(N)$. 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}}\label{begl} \cite{Becker&Glover1971} 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 [[High_codimension_embeddings#Unknotting_theorems|General Position Theorem 2.1]] (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 [[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. Then * (1) $\Sigma(f)$ has a natural orientation. * (2) 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. * (3) the natural orientation on $\Sigma(f)$ extend to $Cl\Sigma(f)$ if $m-n$ is odd \cite{Hudson1969}, Lemma 11.4. * (4) $f\Sigma(f)$ has a natural orientation if $m-n$ is even. Let us prove (1). Take points $x,y\in N$ outside 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 $\eta_y$. (Note 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 $\Sigma(f)$.) Let us prove (4). Take a $(2n-m)$-base $\xi$ at a point $x\in f\Sigma(f)$ outside 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 $(\xi,\eta_+,\eta_-)$. (Note 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$$N \to\Rr^m$ are isotopic. In this page we summarise the situation for $m=2n\ge6$$m=2n\ge6$ and some more general situations.

See general introduction on embeddings, notation and conventions.

2 Classification

Theorem 2.1. Let $N$$N$ be a closed connected $n$$n$-manifold. The Whitney invariant

$\displaystyle W:E^{2n}(N)\to\left\{\begin{array}{cc} H_1(N;\Zz_{(n-1)})& \mbox{either $n$ is odd or $N$ is orientable}\\ \Zz\oplus\Zz_2^{s-1}&\mbox{$n$ is even, $N$ is non-orientable and } H_1(N,\Zz_2)\cong\Zz_2^s\end{array}\right.$ $$\displaystyle W:E^{2n}(N)\to\left\{\begin{array}{cc} H_1(N;\Zz_{(n-1)})& \mbox{either $n$ is odd or $N$ is orientable}\\ \Zz\oplus\Zz_2^{s-1}&\mbox{$n$ is even, $N$ is non-orientable and } H_1(N,\Zz_2)\cong\Zz_2^s\end{array}\right.$$

is bijective if either $n\ge4$$n\ge4$ or $n=3$$n=3$ and CAT=PL.

This is proved in [Haefliger1962b], [Haefliger&Hirsch1963], [Weber1967], [Bausum1975], [Vrabec1977] (a minor miscalculation for the non-orientable case being corrected only in [Vrabec1977]).

The classification of smooth embeddings of 3-manifolds in $\Rr^6$$\Rr^6$ is more complicated.

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

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

3 Examples

Together with the Haefliger knotted sphere, examples of Hudson tori presented below were the first examples of embeddings in codimension greater than 2 not isotopic to the standard embedding. (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 $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$ of standard embeddings.

Let $*:=(-1,0,\ldots,0)\in S^{n-1}$$*:=(-1,0,\ldots,0)\in S^{n-1}$.

3.1 Hudson tori

In this subsection we construct, for $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. (This construction, as opposed to Definition 3.2, works for $n=1$$n=1$.) Take the standard embeddings $2D^{n+1}\times S^{n-1}\subset\Rr^{2n}$$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 embedded sphere and embedded torus

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

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 this is a linked embedded connected sum, i.e. connected sum of two embeddings whose images are not contained in disjoint cubes.)

Definition 3.2. For $a\in\Zz$$a\in\Zz$ instead of $2\partial D^{n+1}\times*$$2\partial D^{n+1}\times*$ we take $|a|$$|a|$ copies $(1+\frac1k)\partial D^{n+1}\times*$$(1+\frac1k)\partial D^{n+1}\times*$ ($k=1,\dots,|a|$$k=1,\dots,|a|$) of $n$$n$-sphere outside $D^{n+1}\times S^{n-1}$$D^{n+1}\times S^{n-1}$ `parallel' to $\partial D^{n+1}\times *$$\partial D^{n+1}\times *$, with 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$$k$-th copy to $(k+1)$$(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')$$\Hud_n(a')$ if and only if $a\equiv a'\mod2$$a\equiv a'\mod2$.

Proposition 3.3 follows by calculation of the Whitney invariant (Remark 4.3.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')$$\Hud_2(a')$ if $a\not\equiv a'\mod2$$a\not\equiv a'\mod2$. 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], \S6).

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 Figure 2.2 of [Skopenkov2006]. 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 by two points lying in $D^n\times x$$D^n\times x$, i.e., by the image of an embedding $S^0\times x\to D^n\times x$$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 Figure 2.3 of [Skopenkov2006].) 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 analogous proof.

(b) 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$$n=2$, or are smoothly isotopic for $n=3$$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 4.3.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$$p=0$ this corresponds to the Zeeman construction and its composition with the second unframed Kirby move. It would be interesting to know if links $\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$$\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 could be further generalized.

3.2 Action by linked embedded connected sum

In this subsection we generalize the construction of 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}$$f_0:N\to\Rr^{2n}$ and $a\in H_1(N)$$a\in H_1(N)$, 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$$n$-sphere representing homology Alexander dual of $a$$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 of $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 certain arc joining their images).

Take a vector field on $S^1\times*$$S^1\times*$ normal to $S^1\times S^{n-1}$$S^1\times S^{n-1}$. Extend $S^1\times*$$S^1\times*$ along this vector field to a map $\overline a:D^2\to\Rr^{2n}$$\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$$\pi_1(V_{2n-2,n-1})=0$. Hence the standard framing of $S^1\times*$$S^1\times*$ 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 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 4.1 of the Whitney invariant, $W(f_a)$$W(f_a)$ is $a$$a$ or $a\mod2$$a\mod2$. Thus

• 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;
• linked embedded connected sum defines a free transitive action of $H_1(N;\Zz_{(n-1)})$$H_1(N;\Zz_{(n-1)})$ on $E^{2n}(N)$$E^{2n}(N)$, unless $n=3$$n=3$ and CAT=DIFF.

Parametric connected sum also defines a free transitive action of $H_1(N;\Zz_{(n-1)})$$H_1(N;\Zz_{(n-1)})$ on $E^{2n}(N)$$E^{2n}(N)$ for $n\ge4$$n\ge4$ [Skopenkov2014], Remark 18.a.

4 The Whitney invariant

Let $N$$N$ be a closed $n$$n$-manifold and $f,f_0:N\to\Rr^m$$f,f_0:N\to\Rr^m$ embeddings. Roughly speaking, $W(f,f_0)=W_{f_0}(f)=W(f)$$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 4.2 in the smooth category, following [Skopenkov2010]. The definition in the PL category is analogous [Hudson1969], \S12, [Vrabec1977], p. 145, [Skopenkov2006], \S2.4. Before we present a simpler Definition 4.1 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 4.1. 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$$N_0$ has an $(m-n-1)$$(m-n-1)$-dimensional spine. Therefor 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

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$\Zz_2$ 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:

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

The orientation on $f\cap F$$f\cap F$ (which extends to $Cl(f\cap F)$$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 this 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$$\Rr^m$, it follows that this condition indeed defines an orientation on $f\cap F$$f\cap F$.

Definition 4.2. 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

Tex syntax error

$\Zz_2$ 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$$m-n$ odd it has a natural orientation and so carries a homology class with $\Zz$$\Zz$ coefficients. Define the Whitney invariant to be the homology class:

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

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 4.3. (a) The Whitney invariant is well-defined by Definition 4.2, i.e. independent of the choice of $H$$H$, analogously to [Skopenkov2006], \S2.4. (The orientation is defined for each $m,n$$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

Tex syntax error

$\Zz_2$-coefficients)

(b) Definition 4.1 is equivalent to Definition 4.2. (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 4.1, 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) Clearly, $W(f)$$W(f)$ is not changed throughout isotopy of $f$$f$. Hence it gives a map $W:E^m(N)\to H_{2n-m+1}(N;\Zz_{(m-n-1)})$$W:E^m(N)\to H_{2n-m+1}(N;\Zz_{(m-n-1)})$.

(d) 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$).

(e) 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$.

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

(e) For $m=2n+1$$m=2n+1$ the Whitney invariant is the collection of pairwise linking coefficients of the components of $N$$N$, cf. Definition 3.2 for $m=2p+1$$m=2p+1$ and $p=q$$p=q$.

5 A generalization to highly-connected manifolds

Let $N$$N$ be a closed $k$$k$-connected $n$$n$-manifold. Recall the unknotting theorem that every two embeddings $N \to\Rr^m$$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 present description of $E^{2n-k}(N)$$E^{2n-k}(N)$ generalizing Theorem 2.1, and its generalization to $E^m(N)$$E^m(N)$ for $m\ge2n-2k+1$$m\ge2n-2k+1$.

Examples are Hudson tori $\Hud_{m,n,p}(a), \Hud'_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$$\Hud_{m,n,p}(a), \Hud'_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$. An action of $H_{k+1}(N;\Z_{(n-k-1)})$$H_{k+1}(N;\Z_{(n-k-1)})$ on $E^{2n-k}(N)$$E^{2n-k}(N)$ for $n\ge2k+4$$n\ge2k+4$ is defined either by analogue of linked embedded connected sum, cf. [Skopenkov2010], Definition 1.4, or by parametric connected sum [Skopenkov2014], Remark 18.a. Both these actions are free transitive by Theorem 5.1 below. If $n=2k+3$$n=2k+3$, then analogue of linked embedded connected sum gives only a construction of embeddings $f_a:N\to\Rr^{2n-k}$$f_a:N\to\Rr^{2n-k}$ for each $a\in H_{k+1}(N;\Z)$$a\in H_{k+1}(N;\Z)$ but not a well-defined action of $H_{k+1}(N;\Z_{(n-k-1)})$$H_{k+1}(N;\Z_{(n-k-1)})$ on $E^{2n-k}(N)$$E^{2n-k}(N)$.

5.1 Classification just below the stable range

Theorem 5.1. Let $N$$N$ be a closed orientable homologically $k$$k$-connected $n$$n$-manifold, $k\ge0$$k\ge0$. Then the Whitney invariant

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

is a bijection, provided $n\ge k+3$$n\ge k+3$ or $n\ge2k+4$$n\ge2k+4$ in the PL or DIFF categories, respectively.

Theorem 5.1 was proved in [Haefliger&Hirsch1963], [Hudson1969], \S11, [Boechat&Haefliger1970], [Boechat1971], [Vrabec1977] homotopically $k$$k$-connected manifolds. The proof works for homologically $k$$k$-connected manifolds.

For $k=0$$k=0$ this is covered by Theorem 2.1; for $k\ge1$$k\ge1$ it is not. The PL case of Theorem 5.1 gives nothing but the Unknotting Spheres Theorem for $k+3\le n\le2k+1$$k+3\le n\le2k+1$.

By Theorem 5.1 the Whitney invariant $W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{(q)}$$W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{(q)}$ is bijective for $1\le p\le q-2$$1\le p\le q-2$. It is in fact a group isomorphism (for the group structure introduced in [Skopenkov2006a], [Skopenkov2015], [Skopenkov2015]). The generator is Hudson torus $\Hud(1)$$\Hud(1)$. Also, for $q\ge2$$q\ge2$ by Theorem 5.1 the Whitney invariants

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

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

5.2 Classification in the presence of smoothly knotted spheres

Because of the existence of knots the analogues of Theorem 5.1 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)$$E^{2n-k}(N)$ is much harder: for 40 years the only known complete readily calculable classification results were for homology sphere $N$$N$. The following result for $n=2k+3$$n=2k+3$ was obtained using the Bo\'echat-Haefliger formula for the smoothing obstruction [Boechat1971].

Theorem 5.2. [Skopenkov2008] Let $N$$N$ be a closed homologically $(2l-2)$$(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 analogue of 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 5.2 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$.

5.3 Classification further below the stable range

How to describe $E^m(N)$$E^m(N)$ for $m<2n-k$$m<2n-k$? See remarks on $E^{2n-1}(N)$$E^{2n-1}(N)$. Some estimations of $E^{2n-k-1}(N)$$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 5.3. [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$$k=0$ this is the same as General Position Theorem 2.1 (because $V_{2n+1,n+1}$$V_{2n+1,n+1}$ is $(n-1)$$(n-1)$-connected). For $k=1$$k=1$ this is covered by Theorem 5.1; for $k\ge2$$k\ge2$ it is not.

E.g. by Theorem 5.3 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 Knotted tori and [Skopenkov2002].

Observe that in Theorem 5.3 $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$.

6 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. Then

• (1) $\Sigma(f)$$\Sigma(f)$ has a natural orientation.
• (2) the natural orientation on $\Sigma(f)$$\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$$S^2\to\Rr^4$ having only one self-intersection point.
• (3) the natural orientation on $\Sigma(f)$$\Sigma(f)$ extend to $Cl\Sigma(f)$$Cl\Sigma(f)$ if $m-n$$m-n$ is odd [Hudson1969], Lemma 11.4.
• (4) $f\Sigma(f)$$f\Sigma(f)$ has a natural orientation if $m-n$$m-n$ is even.

Let us prove (1). Take points $x,y\in N$$x,y\in N$ outside singularities of $\Sigma(f)$$\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$$\eta_y$. (Note that a change of the orientation of $N$$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)$.)

Let us prove (4). Take a $(2n-m)$$(2n-m)$-base $\xi$$\xi$ at a point $x\in f\Sigma(f)$$x\in f\Sigma(f)$ outside singularities of $f\Sigma(f)$$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_-)$$(\xi,\eta_+,\eta_-)$. (Note that a change of the orientation of $N$$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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