Embeddings just below the stable range: classification

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1 Introduction

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

Recall the Whitney-Wu Unknotting Theorem: if $N$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}N$ is a connected manifold of dimension $n>1$$n>1$, and $m \ge2n+1$$m \ge2n+1$, then every two embeddings $N \to\Rr^m$$N \to\Rr^m$ are isotopic [Skopenkov2016c, Theorem 3.2], [Skopenkov2006, Theorem 2.5]. In this page we summarize the situation for $m=2n\ge6$$m=2n\ge6$ and some more general situations.

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

2 Classification

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

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

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

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

is a 1-1 correspondence.

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

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

Remark 2.2 (Comments on the proof). Part (a) is proved in [Haefliger1962b, 1.3.e], [Haefliger1963], [Haefliger&Hirsch1963, Theorem 2.4], [Bausum1975, Theorem 43] in the smooth category, and in [Weber1967], [Vrabec1977, Theorems 1.1 and 1.2] in the PL category.

Part (b) is proved in [Bausum1975, Theorem 43] in the smooth category. According to [Weber1967], [Skopenkov1997] the proof works also in the PL category.

In part (b) we replaced the kernel $\ker Sq^1$$\ker Sq^1$ from [Bausum1975, Theorem 43] by $\Zz_2^{s-1}$$\Zz_2^{s-1}$. This is possible because, as a specialist could see, $Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$$Sq^1:H^{n-1}(N;\Z_2)\to H^n(N;\Z_2)$ is the multiplication with $w_1(N)$$w_1(N)$, so $\ker Sq^1 \cong \Zz_2^{s-1}$$\ker Sq^1 \cong \Zz_2^{s-1}$.

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

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

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

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

3 Hudson tori

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

Take the standard embedding $S^{n-1}\subset\Rr^n\subset\Rr^{2n}$$S^{n-1}\subset\Rr^n\subset\Rr^{2n}$. The natural normal framing on this embedding defines the `standard inclusion'

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

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

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

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

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

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

Define the `standard inclusion'

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

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

Take the embedding $g_1$$g_1$ given by

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

The segment

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$2{\rm in}([1,2]1_n\times1_{n-1})$ joins the images of $\Hud_n(0)$$\Hud_n(0)$ and $g_1$$g_1$; the interior of this segment misses the images. Let $\Hud_n(1)$$\Hud_n(1)$ be the linked embedded connected sum of $\Hud_n(0)$$\Hud_n(0)$ and $g_1$$g_1$ along this segment, compatible with the orientation.

(Here 'linked' means that the images of the embeddings are not contained in disjoint cubes, unlike for the unlinked embedded connected sum [Skopenkov2016c, $\S$$\S$5].)

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

The original motivation for Hudson was that $\Hud_n(1)$$\Hud_n(1)$ is not isotopic to $\Hud_n(0)$$\Hud_n(0)$ for any $n\ge3$$n\ge3$ (this is a particular case of Proposition 3.2 below).

One 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

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

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

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

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

Proposition 3.2 follows by calculation of the Whitney invariant (Remark 5.3.d below) and, for $n$$n$ even, by Theorem 2.1.

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

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

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

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

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$$\displaystyle S^0\times S^{n-1}\overset{\widetilde a\times pr_{S^{n-1}}}\to S^{n-1}\times S^{n-1}\subset D^{n+1}\times S^{n-1}\overset{{\rm in}}\to\Rr^{2n}.$$

See [Skopenkov2006, Figure 2.2]. The image of $\widetilde a\times pr_{S^{n-1}}$$\widetilde a\times pr_{S^{n-1}}$ is the union of the graphs of the maps $\overline a$$\overline a$ and $-\overline a$$-\overline a$.

For any $x\in S^{n-1}$$x\in S^{n-1}$ the disk $D^{n+1}\times x$$D^{n+1}\times x$ intersects the image of this embedding at two points lying in $D^n\times x$$D^n\times x$, i.e., at the image of an embedding $S^0\times x\to D^n\times x$$S^0\times x\to D^n\times x$. The embedding $\Hud_n'(a)$$\Hud_n'(a)$ is obtained by extending the latter embeddings to embeddings $S^1\times x\to D^{n+1}\times x$$S^1\times x\to D^{n+1}\times x$ for all $x$$x$. Cf. [Skopenkov2006, Figure 2.3].

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

(b) The embeddings $\Hud_n(a)$$\Hud_n(a)$ and $\Hud_n'(a)$$\Hud_n'(a)$ are smoothly isotopic for $n\ge4$$n\ge4$ and are PL isotopic for $n\ge3$$n\ge3$ [Skopenkov2006a]. This follows by calculation of the Whitney invariant (Remark 5.3.d below). It would be interesting to know if they are smoothly isotopic for $n=3$$n=3$.

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

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

4 Action by linked embedded connected sum

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

More precisely, represent $a$$a$ by an embedding $a:S^1\to N$$a:S^1\to N$. Since any orientable bundle over $S^1$$S^1$ is trivial, $\nu_{f_0}^{-1}a(S^1)\cong S^1\times S^{n-1}$$\nu_{f_0}^{-1}a(S^1)\cong S^1\times S^{n-1}$. Choose an identification of $\nu_{f_0}^{-1}a(S^1)$$\nu_{f_0}^{-1}a(S^1)$ with $S^1\times S^{n-1}$$S^1\times S^{n-1}$. In the next paragraph we recall definition of embedded surgery on $S^1\times(-1_{n-1})\subset S^1\times S^{n-1}$$S^1\times(-1_{n-1})\subset S^1\times S^{n-1}$, which yields an embedding $g:S^n\to C_{f_0}$$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$$g$, along some arc joining their images. Since $n\ge3$$n\ge3$, the embedding $f_a$$f_a$ is independent of the choises made in the construction, except possibly of the identification $\nu_{f_0}^{-1}a(S^1)=S^1\times S^{n-1}$$\nu_{f_0}^{-1}a(S^1)=S^1\times S^{n-1}$, for which see [Skopenkov2014].

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

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

Take an embedding $g:S^n\to C_{f_0}$$g:S^n\to C_{f_0}$ such that

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

If $n$$n$ is odd, take $g$$g$ representing homology Alexander dual of $a$$a$ (not of $-a$$-a$).

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

Proposition 4.1. For $n\ge3$$n\ge3$ linked embedded connected sum, or parametric connected sum, define free transitive actions of $H_1(N;\Zz_{\varepsilon(n-1)})$$H_1(N;\Zz_{\varepsilon(n-1)})$ on $E^{2n}(N)$$E^{2n}(N)$, unless $n=3$$n=3$ in the smooth category.

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

5 The Whitney invariant

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

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

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

Roughly speaking, $W(f):=[Cl\Sigma(H)]$$W(f):=[Cl\Sigma(H)]$ is defined as the homology class of the closure of the self-intersection set of a general position homotopy $H$$H$ between $f$$f$ and $f_0$$f_0$. This is formalized in Definition 5.2 in the smooth category, following [Skopenkov2010], see also [HaefligerHirsch1963]. The definition in the PL category is analogous [Hudson1969, $\S$$\S$11], [Vrabec1977, p. 145], [Skopenkov2006, $\S$$\S$2.4 `The Whitney invariant']. We begin by presenting a simpler definition, Definition 5.1, for a particular case.

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

Definition 5.1. Assume that $N$$N$ is $(2n-m)$$(2n-m)$-connected and $2m\ge3n+3$$2m\ge3n+3$. Then restrictions of $f$$f$ and $f_0$$f_0$ to $N_0$$N_0$ are regular homotopic (see [Koschorke2013, Definition 2.7], [Hirsch1959]). Since $N$$N$ is $(2n-m)$$(2n-m)$-connected, $N_0$$N_0$ retracts to an $(m-n-1)$$(m-n-1)$-dimensional polyhedron. Therefore these restrictions are isotopic, cf. [Haefliger&Hirsch1963, 3.1.b], [Takase2006, Lemma 2.2]. So we can make an isotopy of $f$$f$ and assume that $f=f_0$$f=f_0$ on $N_0$$N_0$. Take a general position homotopy $F:B^n\times I\to\Rr^m$$F:B^n\times I\to\Rr^m$ relative to $\partial B^n$$\partial B^n$ between the restrictions of $f$$f$ and $f_0$$f_0$ to $B^n$$B^n$. Let $f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$$f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$ (`the intersection of this homotopy with $f(N-B^n)$$f(N-B^n)$'). Since $n+2(n+1)<2m$$n+2(n+1)<2m$, by general position $Cl(f\cap F)$$Cl(f\cap F)$ is a compact $(2n+1-m)$$(2n+1-m)$-manifold whose boundary is contained in $\partial N_0$$\partial N_0$. So $f\cap F$$f\cap F$ carries a homology class with $\Zz_2$$\Zz_2$ coefficients. If $m-n$$m-n$ is odd and $N$$N$ is orientable, $f\cap F$$f\cap F$ has a natural orientation defined below, and so carries a homology class with $\Zz$$\Zz$ coefficients. Define $W(f)$$W(f)$ to be the homology class:

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

The orientation on $f\cap F$$f\cap F$ (extendable to $Cl(f\cap F)$$Cl(f\cap F)$) is defined (for $m-n$$m-n$ odd and $N$$N$ is orientable) as follows. For any point $x\in f\cap F$$x\in f\cap F$ take a vector at $x$$x$ tangent to $f\cap F$$f\cap F$. Complete this vector to a positive base tangent to $N$$N$. Since $n+2(n+1)<2m$$n+2(n+1)<2m$, by general position there is a unique point $y\in B^n\times I$$y\in B^n\times I$ such that $Fy=fx$$Fy=fx$. The tangent vector at $x$$x$ thus gives a tangent vector at $y$$y$ to $B^n\times I$$B^n\times I$. Complete this vector to a positive base tangent to $B^n\times I$$B^n\times I$, where the orientation on $B^n$$B^n$ comes from $N$$N$. The union of the images of the constructed two bases is a base at $Fy=fx$$Fy=fx$ of $\Rr^m$$\Rr^m$. If the latter base is positive, then call the initial vector of $f\cap F$$f\cap F$ 'positive'. Since a change of the orientation on $f\cap F$$f\cap F$ forces a change of the orientation of the latter base of $\Rr^m$$\Rr^m$, this condition indeed defines an orientation on $f\cap F$$f\cap F$.

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

The closure $Cl\Sigma(H)$$Cl\Sigma(H)$ of the self-intersection set carries a cycle mod 2. If $N$$N$ is oriented and $m-n$$m-n$ is odd, the closure also carries an integer cycle. See [Hudson1967, $\S$$\S$11], [Skopenkov2006, $\S$$\S$2.3 `The Whitney obstruction'].

(Let us present informal explanations of these facts. For $2m\ge3n+2$$2m\ge3n+2$ by general position the closure $Cl\Sigma(H)$$Cl\Sigma(H)$ can be assumed to be a submanifold. In general, since $m\ge n+2$$m\ge n+2$, by general position the closure has codimension 2 singularities, see definition in $\S$$\S$7. So the closure carries a cycle mod 2. The closure also has a natural orientation, see Definition 7.1 and remark below. So the closure carries an integer cycle.)

Define the Whitney invariant to be the homology class:

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

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

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

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

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

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

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

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

(c) Since a change of the orientation on $N$$N$ forces a change of the orientation on $B^n$$B^n$, the class $W(f)$$W(f)$ is independent of the choice of the orientation on $N$$N$. For the reflection $\sigma:\Rr^m\to\Rr^m$$\sigma:\Rr^m\to\Rr^m$ with respect to a hyperplane we have $W(\sigma\circ f)=-W(f)$$W(\sigma\circ f)=-W(f)$ (because we may assume that $f=f_0=\sigma\circ f$$f=f_0=\sigma\circ f$ on $N_0$$N_0$ and because a change of the orientation of $\Rr^m$$\Rr^m$ forces a change of the orientation of $f\cap F$$f\cap F$).

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

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

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

This is clear by Definition 5.1 because $W(f\#g)-W(f)=W(f\#g,f_0)-W(f,f_0)=W(f\#g,f)$$W(f\#g)-W(f)=W(f\#g,f_0)-W(f,f_0)=W(f\#g,f)$. Cf. [Skopenkov2008, Addendum to the Classification Theorem].

(f) For $m=2n+1$$m=2n+1$ the Whitney invariant can be easily calculated from (and can be easily refined to obtain) the collection of pairwise linking coefficients of the components of $N$$N$, cf. Remark 3.2.e of [Skopenkov2016h].

(g) The Whitney invariant need not be a bijection for $m<2n$$m<2n$ of for $m=2n$$m=2n$, $n$$n$ even and $N$$N$ non-orientable.

6 A generalization to highly-connected manifolds

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

6.1 Examples

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

Example 6.1 (Linked embedded connected sum, cf. [Skopenkov2010, Definition 1.4]). If $N$$N$ is $k$$k$-connected, then for an embedding $f_0:N\to S^{2n-k}$$f_0:N\to S^{2n-k}$ and a class $a\in H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$$a\in H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$ one can construct an embedding $f_a:N\to S^{2n-k}$$f_a:N\to S^{2n-k}$ by linked connected sum analogously to the case $k=0$$k=0$ presented in \S4.

We have $W(f_a,f_0)=a$$W(f_a,f_0)=a$ for the Whitney invariant. Hence by Theorem 6.2 below this construction gives a free transitive action of $H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$$H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})$ on $E^{2n-k}(N)$$E^{2n-k}(N)$ (provided $n\ge k+3$$n\ge k+3$ or $n\ge2k+4$$n\ge2k+4$ in the PL or smooth categories, respectively). If $n=2k+3$$n=2k+3$, then this construction gives only a construction of embeddings $f_a:N\to\Rr^{2n-k}$$f_a:N\to\Rr^{2n-k}$ for any $a\in H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$$a\in H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$ but not a well-defined action of $H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$$H_{k+1}(N;\Z_{\varepsilon(n-k-1)})$ on $E^{2n-k}(N)$$E^{2n-k}(N)$.

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

6.2 Classification just below the stable range

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

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

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

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

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

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

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

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

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

6.3 Classification in the presence of smoothly knotted spheres

Because of the existence of knotted spheres the analogues of Theorem 6.2 for $n=k+2$$n=k+2$ in the PL case, and for $n\le2k+3$$n\le2k+3$ in the smooth case are false.

So for the smooth category, $n\le2k+3$$n\le2k+3$ and $N$$N$ closed connected, a classification of $E^{2n-k}(N)$$E^{2n-k}(N)$ is much harder: for 40 years the only known complete readily calculable classification results were for homology spheres $N$$N$. E.g.

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

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

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

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

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

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

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

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

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

6.4 Classification further below the stable range

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

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

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

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

E.g. by Theorem 6.4 there is a 1-1 correspondence $E^m(S^p\times S^q)\to\pi_p(V_{m,p+q+1})\oplus\pi_q(V_{m,p+q+1})$$E^m(S^p\times S^q)\to\pi_p(V_{m,p+q+1})\oplus\pi_q(V_{m,p+q+1})$ for $m\ge2q+3$$m\ge2q+3$ and $2m\ge3q+3p+4$$2m\ge3q+3p+4$. For a generalization see [Skopenkov2016k] (to knotted tori) and [Skopenkov2002].

Observe that in Theorem 6.4 $V_{m,n+1}$$V_{m,n+1}$ can be replaced by $V_{M,M+n-m+1}$$V_{M,M+n-m+1}$ for any $M>n$$M>n$.

7 An orientation on the self-intersection set

Let $f:N\to\Rr^m$$f:N\to\Rr^m$ be a general position smooth map of an oriented $n$$n$-manifold $N$$N$. Assume that $m\ge n+2$$m\ge n+2$. Then the closure $Cl\Sigma(f)$$Cl\Sigma(f)$ of the self-intersection set of $f$$f$ has codimension 2 singularities, i.e., there is $X\subset Cl\Sigma(f)$$X\subset Cl\Sigma(f)$ (called a singular set) of dimension at most $\dim Cl\Sigma(f)-2$$\dim Cl\Sigma(f)-2$ such that $Cl\Sigma(f)-X$$Cl\Sigma(f)-X$ is an open manifold.

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

We remark that

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

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

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

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

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

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