Embeddings just below the stable range: classification

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1 Introduction

Recall the unknotting theorem that if $N$$== Introduction == <wikitex>; Recall the [[High codimension embeddings#Unknotting theorems|unknotting theorem]] that if $N$ is a connected manifold of dimension $n>1$, then there is just one isotopy class of embedding $N \to \Rr^m$ if $m \geq 2n + 1$. In this page we summarise the situation for $m = 2n\ge8$, and give references to the case $m=2n-1\ge9$. For notation and conventions see [[High_codimension_embeddings#Notation and conventions|high codimension embeddings]]. </wikitex> == Classification == <wikitex>; {{beginthm|Classification Theorem}}\label{th4} Let $N$ be a closed connected $n$-manifold. The Whitney invariant $$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.$$ is bijective if either $n\ge4$ or $n=3$ and CAT=PL \cite{Haefliger&Hirsch1963}, \cite{Bausum1975}, \cite{Vrabec1977}, cf. \cite{Hudson1969} {{endthm}} The Whitney invariant is defined below. The classification of smooth [[Embeddings of 3-manifolds in 6-space|embeddings of 3-manifolds in 6-space]] is more complicated. For analogous classification of $E^{2n-k}(N)$ see [[Embeddings_of_highly-connected_manifolds|Embeddings of highly-connected manifolds]]. Some estimations of $E^{2n-k-1}(N)$ for a closed $k$-connected $n$-manifold $N$ and $n\ge2k+5$ (including $k=0$) are presented in \cite{Skopenkov2010}. See also [[Embeddings_of_4-manifolds_in_7-space|Embeddings of 4-manifolds in 7-space]]. </wikitex> == Examples == <wikitex>; Together with the Haefliger knotted sphere $S^{2l-1}\to S^{3l}$, Hudson's examples were the first examples of embeddings in codimension greater than 2 not isotopic to the standard embedding. (Hudson's construction \cite{Hudson1963} was not as explicit as the one below). For $m\ge n+2$ define the $standard$ $embedding$ $S^p\times S^{n-p}\to S^m$ as the composition $S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}\to\Rr^m\to S^m$ of standard embeddings. </wikitex> === Hudson tori 1 === <wikitex>; In this subsection, we recall for $a\in\Zz$ and $n\ge2$. Hudson's construction of embedings $$\Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.$$ Take the standard embeddings D^{n+1}\times S^{n-1}\subset\Rr^{2n}$ (where $ means homothety with coefficient 2) and $\partial D^2\subset\partial D^{n+1}$. Fix a point $x\in S^{n-1}$. The Hudson torus $\Hud_n(1)$ is the embedded connected sum of $\partial D^{n+1}\times x\quad\text{with}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset 2D^{n+1} \times S^{n-1}\subset\Rr^{2n}.$$ (Unlike the connected sum mentioned in [[High_codimension_embeddings#Embedded connected sum|embedded conntected sum]] this is a `linked' connected sum, i.e. connected sum of two embeddings whose images are not contained in disjoint cubes.) For $a\in\Zz$ instead of an embedded $n$-sphere \partial D^{n+1}\times x$ we can take $a$ copies $(1+\frac1k)\partial D^{n+1}\times x$ ($k=1,\dots,a$) of $n$-sphere outside $D^{n+1}\times S^{n-1}$ `parallel' to $\partial D^{n+1}\times x$. Then we join these spheres by tubes so that the homotopy class of the resulting embedding $$S^n\to S^{2n}-D^{n+1}\times S^{n-1}\simeq S^{2n}-S^{n-1}\simeq S^n \quad\text{will be}\quad a\in\pi_n(S^n)\cong\Zz.$$ Let $\Hud_n(a)$ be the connected sum of this embedding with the above standard embedding $\partial D^2\times S^{n-1}\subset\Rr^{2n}$. Clearly, $\Hud_n(0)$ is isotopic to the standard embedding. {{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}}. In particular, $\Hud_n(1)$ is not isotopic to $\Hud_n(0)$ for each $n$ (this was the original motivation for Hudson). Proposition \ref{pr3} follows by Remark 5.e and, for $n$ even, by Theorem 4, both results from [[classification just below the stable range]]. It would be interesting to find an explicit construction of an isotopy between $\Hud_{2k}(a)$ and $\Hud_{2k}(a+2)$ (cf. \cite{Vrabec1977}, \S6) and to prove the analogue of Proposition \ref{pr3} for $n=2$. </wikitex> === Hudson tori 2=== <wikitex>; In this subsection we 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$$n>1$, then there is just one isotopy class of embedding $N \to \Rr^m$$N \to \Rr^m$ if $m \geq 2n + 1$$m \geq 2n + 1$. In this page we summarise the situation for $m = 2n\ge8$$m = 2n\ge8$, and give references to the case $m=2n-1\ge9$$m=2n-1\ge9$.

For notation and conventions see high codimension embeddings.

2 Classification

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 [Haefliger&Hirsch1963], [Bausum1975], [Vrabec1977], cf. [Hudson1969]

The Whitney invariant is defined below.

The classification of smooth embeddings of 3-manifolds in 6-space is more complicated. For analogous classification of $E^{2n-k}(N)$$E^{2n-k}(N)$ see Embeddings of highly-connected manifolds. Some estimations of $E^{2n-k-1}(N)$$E^{2n-k-1}(N)$ for a closed $k$$k$-connected $n$$n$-manifold $N$$N$ and $n\ge2k+5$$n\ge2k+5$ (including $k=0$$k=0$) are presented in [Skopenkov2010]. See also Embeddings of 4-manifolds in 7-space.

3 Examples

Together with the Haefliger knotted sphere $S^{2l-1}\to S^{3l}$$S^{2l-1}\to S^{3l}$, Hudson's examples 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 the one below).

For $m\ge n+2$$m\ge n+2$ define the $standard$$standard$ $embedding$$embedding$ $S^p\times S^{n-p}\to S^m$$S^p\times S^{n-p}\to S^m$ as the composition $S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}\to\Rr^m\to S^m$$S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}\to\Rr^m\to S^m$ of standard embeddings.

3.1 Hudson tori 1

In this subsection, we recall for $a\in\Zz$$a\in\Zz$ and $n\ge2$$n\ge2$. Hudson's construction of embedings

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

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}$$\partial D^2\subset\partial D^{n+1}$. Fix a point $x\in S^{n-1}$$x\in S^{n-1}$. The Hudson torus $\Hud_n(1)$$\Hud_n(1)$ is the embedded connected sum of

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

(Unlike the connected sum mentioned in embedded conntected sum this is a `linked' connected sum, i.e. connected sum of two embeddings whose images are not contained in disjoint cubes.)

For $a\in\Zz$$a\in\Zz$ instead of an embedded $n$$n$-sphere $2\partial D^{n+1}\times x$$2\partial D^{n+1}\times x$ we can take $a$$a$ copies $(1+\frac1k)\partial D^{n+1}\times x$$(1+\frac1k)\partial D^{n+1}\times x$ ($k=1,\dots,a$$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 x$$\partial D^{n+1}\times x$. Then we join these spheres by tubes so that the homotopy class of the resulting embedding

$\displaystyle S^n\to S^{2n}-D^{n+1}\times S^{n-1}\simeq S^{2n}-S^{n-1}\simeq S^n \quad\text{will be}\quad a\in\pi_n(S^n)\cong\Zz.$ $$\displaystyle S^n\to S^{2n}-D^{n+1}\times S^{n-1}\simeq S^{2n}-S^{n-1}\simeq S^n \quad\text{will be}\quad a\in\pi_n(S^n)\cong\Zz.$$

Let $\Hud_n(a)$$\Hud_n(a)$ be the connected sum of this embedding with the above standard embedding $\partial D^2\times S^{n-1}\subset\Rr^{2n}$$\partial D^2\times S^{n-1}\subset\Rr^{2n}$.

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

Proposition 3.1. 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$.

.

In particular, $\Hud_n(1)$$\Hud_n(1)$ is not isotopic to $\Hud_n(0)$$\Hud_n(0)$ for each $n$$n$ (this was the original motivation for Hudson).

Proposition 3.1 follows by Remark 5.e and, for $n$$n$ even, by Theorem 4, both results from classification just below the stable range.

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], \S6) and to prove the analogue of Proposition 3.1 for $n=2$$n=2$.

3.2 Hudson tori 2

In this subsection we 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}$$1\times S^{n-1}$ and the standard embedding $\varphi:\{-1\}\times S^{n-1}\to\partial D^n\subset D^n$$\varphi:\{-1\}\times S^{n-1}\to\partial D^n\subset D^n$ on the other component. This map gives an $embedding$$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].) Each 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$. Extend this embedding $S^0\to D^n\times x$$S^0\to D^n\times x$ for each $x\in S^{n-1}$$x\in S^{n-1}$ to an embedding $S^1\to D^{n+1}\times x$$S^1\to D^{n+1}\times x$. (See Figure 2.3 of [Skopenkov2006].) Thus we obtain the Hudson torus

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

The embedding $\Hud_n'(a)$$\Hud_n'(a)$ is obtained in the same way starting from a map $\varphi$$\varphi$ of degree $a$$a$.

The same proposition as above holds with $\Hud_n$$\Hud_n$ replaced to $\Hud_n'$$\Hud_n'$.

3.3 Remarks

We have $\Hud_n(a)$$\Hud_n(a)$ is PL isotopic to $\Hud_n'(a)$$\Hud_n'(a)$ [Skopenkov2006a]. It would be interesting to prove the smooth analogue of this result.

For $n=1$$n=1$ these construction give what we call the $left$$left$ Hudson torus. The $right$$right$ Hudson torus is constructed analogously and is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of $S^1\times S^1$$S^1\times S^1$.

Analogously one constructs the Hudson torus $\Hud_{4,2}(a):S^2\times S^2\to\Rr^7$$\Hud_{4,2}(a):S^2\times S^2\to\Rr^7$ for $a\in\Zz$$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})$ and $\Hud_{m,n,p}'(a):S^p\times S^{n-p}\to\Rr^m$$\Hud_{m,n,p}'(a):S^p\times S^{n-p}\to\Rr^m$ for $a\in\pi_{n-p}(S^{m-n-1})$$a\in\pi_{n-p}(S^{m-n-1})$.

3.4 An action of the first homology group on embeddings

In this subsection, for $a\in H_1(N)$$a\in H_1(N)$ and for orientable $N$$N$ with $n\ge3$$n\ge3$, we construct an embedding $f_a:N\to\Rr^{2n}$$f_a:N\to\Rr^{2n}$ from an embedding $f_0:N\to\Rr^{2n}$$f_0:N\to\Rr^{2n}$.

For $a \in H_1(N)$$a \in H_1(N)$, represent $a$$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}$$S^1\times S^{n-1}$. It remains to make an embedded surgery of $S^1\times*\subset S^1\times S^{n-1}$$S^1\times*\subset S^1\times S^{n-1}$ to obtain an $n$$n$-sphere $\Sigma\subset C_{f_0}$$\Sigma\subset C_{f_0}$, and then we set $f_a:=f_0\# \Sigma$$f_a:=f_0\# \Sigma$.

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*$$S^1\times*$ along this vector field to a smooth map $\overline a:D^2\to S^{2n}$$\overline a:D^2\to S^{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

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

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

This construction generalizes the construction of $\Hud_n(a)$$\Hud_n(a)$ (from $\Hud_n(0)$$\Hud_n(0)$).

Clearly, $W(f_a)$$W(f_a)$ is $a$$a$ or $a\mod2$$a\mod2$. Thus unless $n=3$$n=3$ and CAT=DIFF

• all isotopy classes of embedings $N\to\Rr^{2n}$$N\to\Rr^{2n}$ can be obtained (from a certain given embedding $f_0$$f_0$) by the above construction;
• the above construction defines an action $H_1(N;\Zz_{(n-1)})\to E^{2n}(N)$$H_1(N;\Zz_{(n-1)})\to E^{2n}(N)$.

4 The Whitney invariant (for either n odd or N orientable)

Fix orientations on $\Rr^{2n}$$\Rr^{2n}$ and, if $N$$N$ is even, on $N$$N$. Fix an embedding $f_0:N\to\Rr^{2n}$$f_0:N\to\Rr^{2n}$. For an embedding $f:N\to\Rr^{2n}$$f:N\to\Rr^{2n}$ the restrictions of $f$$f$ and $f_0$$f_0$ to $N_0$$N_0$ are regular homotopic [Hirsch1959]. Since $N_0$$N_0$ has an $(n-1)$$(n-1)$-dimensional spine, it follows that 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$$N_0$. Take a general position homotopy $F:B^n\times I\to\Rr^{2n}$$F:B^n\times I\to\Rr^{2n}$ relative to $\partial B^n$$\partial B^n$ between the restrictions of $f$$f$ and $g$$g$ to $B^n$$B^n$. Then $f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$$f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$ (i.e. `the intersection of this homotopy with $f(N-B^n)$$f(N-B^n)$') is a 1-manifold (possibly non-compact) without boundary. Define $W(f)$$W(f)$ to be the homology class of the closure of this 1-manifold:

$\displaystyle W(f):=[Cl(f\cap F)]\in H_1(N_0,\partial N_0;\Zz_{(n)})\cong H_1(N;\Zz_{(n)}).$ $$\displaystyle W(f):=[Cl(f\cap F)]\in H_1(N_0,\partial N_0;\Zz_{(n)})\cong H_1(N;\Zz_{(n)}).$$

The orientation on $f\cap F$$f\cap F$ is defined for $N$$N$ orientable as follows. (This orientation is defined for each $n$$n$ but used only for odd $n$$n$.) 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)>2\cdot2n$$n+2(n+1)>2\cdot2n$, by general position there is a unique point $y\in B^n\times I$$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^{2n}$$\Rr^{2n}$. If this base is positive, then call the initial vector of $f\cap F$$f\cap F$ positive. Since a change of the orientation on $f\cap F$$f\cap F$ forces a change of the orientation of the latter base of $\Rr^{2n}$$\Rr^{2n}$, it follows that this condition indeed defines an orientation on $f\cap F$$f\cap F$.

Remark 4.1.

• The Whitney invariant is well-defined, i.e. independent of the choice of $F$$F$ and of the isotopy making $f=f_0$$f=f_0$ outside $B^n$$B^n$. This is so because the above definition is clearly equivalent to the following: $W(f)$$W(f)$ is the homology class of the algebraic sum of the top-dimensional simplices of the self-intersection set $\Sigma(H):=Cl\{x\in N\times I\ |\ \#H^{-1}Hx>1\}$$\Sigma(H):=Cl\{x\in N\times I\ |\ \#H^{-1}Hx>1\}$ of a general position homotopy $H$$H$ between $f$$f$ and $f_0$$f_0$. (For details and definition of the signs of the simplices see [Hudson1969], \S12, [Vrabec1977], p. 145, [Skopenkov2006], \S2, [Skopenkov2010].) It is for being well-defined that we need $\Zz_2$$\Zz_2$-coefficients when $n$$n$ is even.
• Clearly, $W(f_0)=0$$W(f_0)=0$. The definition of $W$$W$ depends on the choice of $f_0$$f_0$, but we write $W$$W$ not $W_{f_0}$$W_{f_0}$ for brevity.
• Since a change of the orientation on $N$$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$$N$. For the reflection $\sigma:\Rr^{2n}\to\Rr^{2n}$$\sigma:\Rr^{2n}\to\Rr^{2n}$ 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$$N_0$ and because a change of the orientation of $\Rr^{2n}$$\Rr^{2n}$ forces a change of the orientation of $f\cap F$$f\cap F$).
• The above definition makes sense for each $n$$n$, not only for $n\ge3$$n\ge3$.
• Clearly, $W(\Hud_n(a))$$W(\Hud_n(a))$ is $a$$a$ or $a\mod2$$a\mod2$ for $n\ge2$$n\ge2$ for the Hudson tori.
• $W(f\#g)=W(f)$$W(f\#g)=W(f)$ for each embeddings $f:N\to\Rr^{2n}$$f:N\to\Rr^{2n}$ and $g:S^n\to\Rr^{2n}$$g:S^n\to\Rr^{2n}$.

5 A generalization to highly-connected manifolds

Examples are the above 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$. See also [Milgram&Rees1971].

5.1 Classification

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$$homotopically$ $k$$k$-connected manifolds. The proof works for $homologically$$homologically$ $k$$k$-connected manifolds.

E.g. by Theorem 5.1 we obtain that 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; the generator is the Hudson torus.

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$.

Analogously to Theorem 5.1 it may be proved that

• if $N$$N$ is a closed connected non-orientable $n$$n$-manifold, then

$\displaystyle E^{2n}(N)=\begin{cases} H_1(N,\Zz_2)& n\text{ odd, }\\ \Zz\oplus\Zz_2^{s-1}&n\text{ even and }H_1(N,\Zz_2)\cong\Zz_2^s\end{cases},$ $$\displaystyle E^{2n}(N)=\begin{cases} H_1(N,\Zz_2)& n\text{ odd, }\\ \Zz\oplus\Zz_2^{s-1}&n\text{ even and }H_1(N,\Zz_2)\cong\Zz_2^s\end{cases},$$

provided $n\ge3$$n\ge3$ or $n\ge4$$n\ge4$ in the PL or DIFF categories, respectively.

This is proved in [Haefliger1962b], [Weber1967], [Bausum1975], [Vrabec1977] (a minor miscalculation being corrected only in [Vrabec1977]).

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 and $n\le2k+3$$n\le2k+3$ a classification is much harder: for 40 years the $only$$only$ known concrete complete classification results were for spheres. The following result was obtained using the Kreck modification of surgery theory.

Theorem 5.2. [Skopenkov2006] Let $N$$N$ be a closed homologically $(2k-2)$$(2k-2)$-connected $(4k-1)$$(4k-1)$-manifold. Then the Whitney invariant $W:E^{6k}_D(N)\to H_{2k-1}(N)$$W:E^{6k}_D(N)\to H_{2k-1}(N)$ is surjective and for each $u\in H_{2k-1}(N)$$u\in H_{2k-1}(N)$ there is a 1--1 correspondence $\eta_u:W^{-1}u\to\Zz_{d(u)}$$\eta_u:W^{-1}u\to\Zz_{d(u)}$, 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 we obtain that

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

6 References

• [Bausum1975] D. R. Bausum, Embeddings and immersions of manifolds in Euclidean space, Trans. Amer. Math. Soc. 213 (1975), 263–303. MR0474330 (57 #13976) Zbl 0323.57017
• [Boechat&Haefliger1970] J. Boéchat and A. Haefliger, Plongements différentiables des variétés orientées de dimension $4$$4$ dans $\Rr^7$$\Rr^7$, (French) Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York (1970), 156–166. MR0270384 (42 #5273) Zbl 0199.27003
• [Boechat1971] J. Boéchat, Plongements de variétées différentiables orientées de dimension $4k$$4k$ dans $\Rr^{6k+1}$$\Rr^{6k+1}$, Comment. Math. Helv. 46 (1971), 141–161. MR0295373 (45 #4439) Zbl 0218.57016
• [Haefliger&Hirsch1963] A. Haefliger and M. W. Hirsch, On the existence and classification of differentiable embeddings, Topology 2 (1963), 129–135. see also MR0149494 (26 #6981) Zbl 0113.38607
• [Haefliger1962b] A. Haefliger, Plongements de variétés dans le domain stable, Séminaire Bourbaki, 245 (1962).
• [Hirsch1959] M. W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR0119214 (22 #9980) Zbl 0118.18603
• [Hudson1963] J. F. P. Hudson, Knotted tori, Topology 2 (1963), 11–22. MR0146829 (26 #4349) Zbl 0151.32503
• [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
• [Milgram&Rees1971] R. Milgram and E. Rees, On the normal bundle to an embedding., Topology 10 (1971), 299-308. MR0290391 (44 #7572) Zbl 0207.22302
• [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
• [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
• [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into $\Rr^{2n-k-1}$$\Rr^{2n-k-1}$, Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
• [Takase2006] M. Takase, Homology 3-spheres in codimension three, Internat. J. Math. 17 (2006), no.8, 869–885.

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

• [Vrabec1977] J. Vrabec, Knotting a $k$$k$-connected closed $\PL$$\PL$ $m$$m$-manifold in $E^{2m-k}$$E^{2m-k}$, Trans. Amer. Math. Soc. 233 (1977), 137–165. MR0645405 (58 #31097) Zbl 386.57013
• [Weber1967] C. Weber, Plongements de polyedres dans le domaine metastable, Comment. Math. Helv. 42 (1967), 1-27. MR0238330 (38 #6606) Zbl 0152.22402

This page has not been refereed. The information given here might be incomplete or provisional.

\in D^n$ on one component N is a connected manifold of dimension $n>1$$n>1$, then there is just one isotopy class of embedding $N \to \Rr^m$$N \to \Rr^m$ if $m \geq 2n + 1$$m \geq 2n + 1$. In this page we summarise the situation for $m = 2n\ge8$$m = 2n\ge8$, and give references to the case $m=2n-1\ge9$$m=2n-1\ge9$.

For notation and conventions see high codimension embeddings.

2 Classification

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 [Haefliger&Hirsch1963], [Bausum1975], [Vrabec1977], cf. [Hudson1969]

The Whitney invariant is defined below.

The classification of smooth embeddings of 3-manifolds in 6-space is more complicated. For analogous classification of $E^{2n-k}(N)$$E^{2n-k}(N)$ see Embeddings of highly-connected manifolds. Some estimations of $E^{2n-k-1}(N)$$E^{2n-k-1}(N)$ for a closed $k$$k$-connected $n$$n$-manifold $N$$N$ and $n\ge2k+5$$n\ge2k+5$ (including $k=0$$k=0$) are presented in [Skopenkov2010]. See also Embeddings of 4-manifolds in 7-space.

3 Examples

Together with the Haefliger knotted sphere $S^{2l-1}\to S^{3l}$$S^{2l-1}\to S^{3l}$, Hudson's examples 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 the one below).

For $m\ge n+2$$m\ge n+2$ define the $standard$$standard$ $embedding$$embedding$ $S^p\times S^{n-p}\to S^m$$S^p\times S^{n-p}\to S^m$ as the composition $S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}\to\Rr^m\to S^m$$S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}\to\Rr^m\to S^m$ of standard embeddings.

3.1 Hudson tori 1

In this subsection, we recall for $a\in\Zz$$a\in\Zz$ and $n\ge2$$n\ge2$. Hudson's construction of embedings

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

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}$$\partial D^2\subset\partial D^{n+1}$. Fix a point $x\in S^{n-1}$$x\in S^{n-1}$. The Hudson torus $\Hud_n(1)$$\Hud_n(1)$ is the embedded connected sum of

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

(Unlike the connected sum mentioned in embedded conntected sum this is a `linked' connected sum, i.e. connected sum of two embeddings whose images are not contained in disjoint cubes.)

For $a\in\Zz$$a\in\Zz$ instead of an embedded $n$$n$-sphere $2\partial D^{n+1}\times x$$2\partial D^{n+1}\times x$ we can take $a$$a$ copies $(1+\frac1k)\partial D^{n+1}\times x$$(1+\frac1k)\partial D^{n+1}\times x$ ($k=1,\dots,a$$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 x$$\partial D^{n+1}\times x$. Then we join these spheres by tubes so that the homotopy class of the resulting embedding

$\displaystyle S^n\to S^{2n}-D^{n+1}\times S^{n-1}\simeq S^{2n}-S^{n-1}\simeq S^n \quad\text{will be}\quad a\in\pi_n(S^n)\cong\Zz.$ $$\displaystyle S^n\to S^{2n}-D^{n+1}\times S^{n-1}\simeq S^{2n}-S^{n-1}\simeq S^n \quad\text{will be}\quad a\in\pi_n(S^n)\cong\Zz.$$

Let $\Hud_n(a)$$\Hud_n(a)$ be the connected sum of this embedding with the above standard embedding $\partial D^2\times S^{n-1}\subset\Rr^{2n}$$\partial D^2\times S^{n-1}\subset\Rr^{2n}$.

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

Proposition 3.1. 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$.

.

In particular, $\Hud_n(1)$$\Hud_n(1)$ is not isotopic to $\Hud_n(0)$$\Hud_n(0)$ for each $n$$n$ (this was the original motivation for Hudson).

Proposition 3.1 follows by Remark 5.e and, for $n$$n$ even, by Theorem 4, both results from classification just below the stable range.

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], \S6) and to prove the analogue of Proposition 3.1 for $n=2$$n=2$.

3.2 Hudson tori 2

In this subsection we 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}$$1\times S^{n-1}$ and the standard embedding $\varphi:\{-1\}\times S^{n-1}\to\partial D^n\subset D^n$$\varphi:\{-1\}\times S^{n-1}\to\partial D^n\subset D^n$ on the other component. This map gives an $embedding$$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].) Each 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$. Extend this embedding $S^0\to D^n\times x$$S^0\to D^n\times x$ for each $x\in S^{n-1}$$x\in S^{n-1}$ to an embedding $S^1\to D^{n+1}\times x$$S^1\to D^{n+1}\times x$. (See Figure 2.3 of [Skopenkov2006].) Thus we obtain the Hudson torus

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

The embedding $\Hud_n'(a)$$\Hud_n'(a)$ is obtained in the same way starting from a map $\varphi$$\varphi$ of degree $a$$a$.

The same proposition as above holds with $\Hud_n$$\Hud_n$ replaced to $\Hud_n'$$\Hud_n'$.

3.3 Remarks

We have $\Hud_n(a)$$\Hud_n(a)$ is PL isotopic to $\Hud_n'(a)$$\Hud_n'(a)$ [Skopenkov2006a]. It would be interesting to prove the smooth analogue of this result.

For $n=1$$n=1$ these construction give what we call the $left$$left$ Hudson torus. The $right$$right$ Hudson torus is constructed analogously and is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of $S^1\times S^1$$S^1\times S^1$.

Analogously one constructs the Hudson torus $\Hud_{4,2}(a):S^2\times S^2\to\Rr^7$$\Hud_{4,2}(a):S^2\times S^2\to\Rr^7$ for $a\in\Zz$$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})$ and $\Hud_{m,n,p}'(a):S^p\times S^{n-p}\to\Rr^m$$\Hud_{m,n,p}'(a):S^p\times S^{n-p}\to\Rr^m$ for $a\in\pi_{n-p}(S^{m-n-1})$$a\in\pi_{n-p}(S^{m-n-1})$.

3.4 An action of the first homology group on embeddings

In this subsection, for $a\in H_1(N)$$a\in H_1(N)$ and for orientable $N$$N$ with $n\ge3$$n\ge3$, we construct an embedding $f_a:N\to\Rr^{2n}$$f_a:N\to\Rr^{2n}$ from an embedding $f_0:N\to\Rr^{2n}$$f_0:N\to\Rr^{2n}$.

For $a \in H_1(N)$$a \in H_1(N)$, represent $a$$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}$$S^1\times S^{n-1}$. It remains to make an embedded surgery of $S^1\times*\subset S^1\times S^{n-1}$$S^1\times*\subset S^1\times S^{n-1}$ to obtain an $n$$n$-sphere $\Sigma\subset C_{f_0}$$\Sigma\subset C_{f_0}$, and then we set $f_a:=f_0\# \Sigma$$f_a:=f_0\# \Sigma$.

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*$$S^1\times*$ along this vector field to a smooth map $\overline a:D^2\to S^{2n}$$\overline a:D^2\to S^{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

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

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

This construction generalizes the construction of $\Hud_n(a)$$\Hud_n(a)$ (from $\Hud_n(0)$$\Hud_n(0)$).

Clearly, $W(f_a)$$W(f_a)$ is $a$$a$ or $a\mod2$$a\mod2$. Thus unless $n=3$$n=3$ and CAT=DIFF

• all isotopy classes of embedings $N\to\Rr^{2n}$$N\to\Rr^{2n}$ can be obtained (from a certain given embedding $f_0$$f_0$) by the above construction;
• the above construction defines an action $H_1(N;\Zz_{(n-1)})\to E^{2n}(N)$$H_1(N;\Zz_{(n-1)})\to E^{2n}(N)$.

4 The Whitney invariant (for either n odd or N orientable)

Fix orientations on $\Rr^{2n}$$\Rr^{2n}$ and, if $N$$N$ is even, on $N$$N$. Fix an embedding $f_0:N\to\Rr^{2n}$$f_0:N\to\Rr^{2n}$. For an embedding $f:N\to\Rr^{2n}$$f:N\to\Rr^{2n}$ the restrictions of $f$$f$ and $f_0$$f_0$ to $N_0$$N_0$ are regular homotopic [Hirsch1959]. Since $N_0$$N_0$ has an $(n-1)$$(n-1)$-dimensional spine, it follows that 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$$N_0$. Take a general position homotopy $F:B^n\times I\to\Rr^{2n}$$F:B^n\times I\to\Rr^{2n}$ relative to $\partial B^n$$\partial B^n$ between the restrictions of $f$$f$ and $g$$g$ to $B^n$$B^n$. Then $f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$$f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$ (i.e. `the intersection of this homotopy with $f(N-B^n)$$f(N-B^n)$') is a 1-manifold (possibly non-compact) without boundary. Define $W(f)$$W(f)$ to be the homology class of the closure of this 1-manifold:

$\displaystyle W(f):=[Cl(f\cap F)]\in H_1(N_0,\partial N_0;\Zz_{(n)})\cong H_1(N;\Zz_{(n)}).$ $$\displaystyle W(f):=[Cl(f\cap F)]\in H_1(N_0,\partial N_0;\Zz_{(n)})\cong H_1(N;\Zz_{(n)}).$$

The orientation on $f\cap F$$f\cap F$ is defined for $N$$N$ orientable as follows. (This orientation is defined for each $n$$n$ but used only for odd $n$$n$.) 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)>2\cdot2n$$n+2(n+1)>2\cdot2n$, by general position there is a unique point $y\in B^n\times I$$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^{2n}$$\Rr^{2n}$. If this base is positive, then call the initial vector of $f\cap F$$f\cap F$ positive. Since a change of the orientation on $f\cap F$$f\cap F$ forces a change of the orientation of the latter base of $\Rr^{2n}$$\Rr^{2n}$, it follows that this condition indeed defines an orientation on $f\cap F$$f\cap F$.

Remark 4.1.

• The Whitney invariant is well-defined, i.e. independent of the choice of $F$$F$ and of the isotopy making $f=f_0$$f=f_0$ outside $B^n$$B^n$. This is so because the above definition is clearly equivalent to the following: $W(f)$$W(f)$ is the homology class of the algebraic sum of the top-dimensional simplices of the self-intersection set $\Sigma(H):=Cl\{x\in N\times I\ |\ \#H^{-1}Hx>1\}$$\Sigma(H):=Cl\{x\in N\times I\ |\ \#H^{-1}Hx>1\}$ of a general position homotopy $H$$H$ between $f$$f$ and $f_0$$f_0$. (For details and definition of the signs of the simplices see [Hudson1969], \S12, [Vrabec1977], p. 145, [Skopenkov2006], \S2, [Skopenkov2010].) It is for being well-defined that we need $\Zz_2$$\Zz_2$-coefficients when $n$$n$ is even.
• Clearly, $W(f_0)=0$$W(f_0)=0$. The definition of $W$$W$ depends on the choice of $f_0$$f_0$, but we write $W$$W$ not $W_{f_0}$$W_{f_0}$ for brevity.
• Since a change of the orientation on $N$$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$$N$. For the reflection $\sigma:\Rr^{2n}\to\Rr^{2n}$$\sigma:\Rr^{2n}\to\Rr^{2n}$ 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$$N_0$ and because a change of the orientation of $\Rr^{2n}$$\Rr^{2n}$ forces a change of the orientation of $f\cap F$$f\cap F$).
• The above definition makes sense for each $n$$n$, not only for $n\ge3$$n\ge3$.
• Clearly, $W(\Hud_n(a))$$W(\Hud_n(a))$ is $a$$a$ or $a\mod2$$a\mod2$ for $n\ge2$$n\ge2$ for the Hudson tori.
• $W(f\#g)=W(f)$$W(f\#g)=W(f)$ for each embeddings $f:N\to\Rr^{2n}$$f:N\to\Rr^{2n}$ and $g:S^n\to\Rr^{2n}$$g:S^n\to\Rr^{2n}$.

5 A generalization to highly-connected manifolds

Examples are the above 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$. See also [Milgram&Rees1971].

5.1 Classification

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$$homotopically$ $k$$k$-connected manifolds. The proof works for $homologically$$homologically$ $k$$k$-connected manifolds.

E.g. by Theorem 5.1 we obtain that 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; the generator is the Hudson torus.

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$.

Analogously to Theorem 5.1 it may be proved that

• if $N$$N$ is a closed connected non-orientable $n$$n$-manifold, then

$\displaystyle E^{2n}(N)=\begin{cases} H_1(N,\Zz_2)& n\text{ odd, }\\ \Zz\oplus\Zz_2^{s-1}&n\text{ even and }H_1(N,\Zz_2)\cong\Zz_2^s\end{cases},$ $$\displaystyle E^{2n}(N)=\begin{cases} H_1(N,\Zz_2)& n\text{ odd, }\\ \Zz\oplus\Zz_2^{s-1}&n\text{ even and }H_1(N,\Zz_2)\cong\Zz_2^s\end{cases},$$

provided $n\ge3$$n\ge3$ or $n\ge4$$n\ge4$ in the PL or DIFF categories, respectively.

This is proved in [Haefliger1962b], [Weber1967], [Bausum1975], [Vrabec1977] (a minor miscalculation being corrected only in [Vrabec1977]).

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 and $n\le2k+3$$n\le2k+3$ a classification is much harder: for 40 years the $only$$only$ known concrete complete classification results were for spheres. The following result was obtained using the Kreck modification of surgery theory.

Theorem 5.2. [Skopenkov2006] Let $N$$N$ be a closed homologically $(2k-2)$$(2k-2)$-connected $(4k-1)$$(4k-1)$-manifold. Then the Whitney invariant $W:E^{6k}_D(N)\to H_{2k-1}(N)$$W:E^{6k}_D(N)\to H_{2k-1}(N)$ is surjective and for each $u\in H_{2k-1}(N)$$u\in H_{2k-1}(N)$ there is a 1--1 correspondence $\eta_u:W^{-1}u\to\Zz_{d(u)}$$\eta_u:W^{-1}u\to\Zz_{d(u)}$, 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 we obtain that

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

6 References

• [Bausum1975] D. R. Bausum, Embeddings and immersions of manifolds in Euclidean space, Trans. Amer. Math. Soc. 213 (1975), 263–303. MR0474330 (57 #13976) Zbl 0323.57017
• [Boechat&Haefliger1970] J. Boéchat and A. Haefliger, Plongements différentiables des variétés orientées de dimension $4$$4$ dans $\Rr^7$$\Rr^7$, (French) Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York (1970), 156–166. MR0270384 (42 #5273) Zbl 0199.27003
• [Boechat1971] J. Boéchat, Plongements de variétées différentiables orientées de dimension $4k$$4k$ dans $\Rr^{6k+1}$$\Rr^{6k+1}$, Comment. Math. Helv. 46 (1971), 141–161. MR0295373 (45 #4439) Zbl 0218.57016
• [Haefliger&Hirsch1963] A. Haefliger and M. W. Hirsch, On the existence and classification of differentiable embeddings, Topology 2 (1963), 129–135. see also MR0149494 (26 #6981) Zbl 0113.38607
• [Haefliger1962b] A. Haefliger, Plongements de variétés dans le domain stable, Séminaire Bourbaki, 245 (1962).
• [Hirsch1959] M. W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR0119214 (22 #9980) Zbl 0118.18603
• [Hudson1963] J. F. P. Hudson, Knotted tori, Topology 2 (1963), 11–22. MR0146829 (26 #4349) Zbl 0151.32503
• [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
• [Milgram&Rees1971] R. Milgram and E. Rees, On the normal bundle to an embedding., Topology 10 (1971), 299-308. MR0290391 (44 #7572) Zbl 0207.22302
• [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
• [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
• [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into $\Rr^{2n-k-1}$$\Rr^{2n-k-1}$, Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
• [Takase2006] M. Takase, Homology 3-spheres in codimension three, Internat. J. Math. 17 (2006), no.8, 869–885.

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

• [Vrabec1977] J. Vrabec, Knotting a $k$$k$-connected closed $\PL$$\PL$ $m$$m$-manifold in $E^{2m-k}$$E^{2m-k}$, Trans. Amer. Math. Soc. 233 (1977), 137–165. MR0645405 (58 #31097) Zbl 386.57013
• [Weber1967] C. Weber, Plongements de polyedres dans le domaine metastable, Comment. Math. Helv. 42 (1967), 1-27. MR0238330 (38 #6606) Zbl 0152.22402

This page has not been refereed. The information given here might be incomplete or provisional.

\times S^{n-1}$ and the standard embedding $\varphi:\{-1\}\times S^{n-1}\to\partial D^n\subset D^n$ 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}.) Each disk $D^{n+1}\times x$ intersects the image of this embedding at two points lying in $D^n\times x$. Extend this embedding $S^0\to D^n\times x$ for each $x\in S^{n-1}$ to an embedding $S^1\to D^{n+1}\times x$. (See Figure 2.3 of \cite{Skopenkov2006}.) Thus we obtain the Hudson torus $$\Hud_n'(1):S^1\times S^{n-1}\to D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$ The embedding $\Hud_n'(a)$ is obtained in the same way starting from a map $\varphi$ of degree $a$. The same proposition as above holds with $\Hud_n$ replaced to $\Hud_n'$. === Remarks === ; We have $\Hud_n(a)$ is PL isotopic to $\Hud_n'(a)$ \cite{Skopenkov2006a}. It would be interesting to prove the smooth analogue of this result. For $n=1$ these construction give what we call the $left$ Hudson torus. The $right$ Hudson torus is constructed analogously and is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of $S^1\times S^1$. Analogously one constructs the Hudson torus $\Hud_{4,2}(a):S^2\times S^2\to\Rr^7$ for $a\in\Zz$ or, more generally, $\Hud_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$ for $a\in\pi_n(S^{m-n+p-1})$ and $\Hud_{m,n,p}'(a):S^p\times S^{n-p}\to\Rr^m$ for $a\in\pi_{n-p}(S^{m-n-1})$. === An action of the first homology group on embeddings === ; In this subsection, for $a\in H_1(N)$ and for orientable $N$ with $n\ge3$, we construct an embedding $f_a:N\to\Rr^{2n}$ from an embedding $f_0:N\to\Rr^{2n}$. For $a \in H_1(N)$, 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 $S^1\times S^{n-1}$. It remains to make an embedded surgery of $S^1\times*\subset S^1\times S^{n-1}$ to obtain an $n$-sphere $\Sigma\subset C_{f_0}$, and then we set $f_a:=f_0\# \Sigma$. 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 smooth map $\overline a:D^2\to S^{2n}$. 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 $$\widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}$$ $$\mbox{Let}\qquad \Sigma:\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n.$$ This construction generalizes the construction of $\Hud_n(a)$ (from $\Hud_n(0)$). Clearly, $W(f_a)$ is $a$ or $a\mod2$. Thus unless $n=3$ and CAT=DIFF * all isotopy classes of embedings $N\to\Rr^{2n}$ can be obtained (from a certain given embedding $f_0$) by the above construction; * the above construction defines an action $H_1(N;\Zz_{(n-1)})\to E^{2n}(N)$. == The Whitney invariant (for either n odd or N orientable) == ; Fix orientations on $\Rr^{2n}$ and, if $N$ is even, on $N$. Fix an embedding $f_0:N\to\Rr^{2n}$. For an embedding $f:N\to\Rr^{2n}$ the restrictions of $f$ and $f_0$ to $N_0$ are regular homotopic \cite{Hirsch1959}. Since $N_0$ has an $(n-1)$-dimensional spine, it follows that 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 $F:B^n\times I\to\Rr^{2n}$ relative to $\partial B^n$ between the restrictions of $f$ and $g$ to $B^n$. Then $f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$ (i.e. `the intersection of this homotopy with $f(N-B^n)$') is a 1-manifold (possibly non-compact) without boundary. Define $W(f)$ to be the homology class of the closure of this 1-manifold: $$W(f):=[Cl(f\cap F)]\in H_1(N_0,\partial N_0;\Zz_{(n)})\cong H_1(N;\Zz_{(n)}).$$ The orientation on $f\cap F$ is defined for $N$ orientable as follows. (This orientation is defined for each $n$ but used only for odd $n$.) 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 $n+2(n+1)>2\cdot2n$, 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 $\Rr^{2n}$. If this 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 $\Rr^{2n}$, it follows that this condition indeed defines an orientation on $f\cap F$. {{beginthm|Remark}}\label{re5} *The Whitney invariant is well-defined, i.e. independent of the choice of $F$ and of the isotopy making $f=f_0$ outside $B^n$. This is so because the above definition is clearly equivalent to the following: $W(f)$ is the homology class of the algebraic sum of the top-dimensional simplices of the self-intersection set $\Sigma(H):=Cl\{x\in N\times I\ |\ \#H^{-1}Hx>1\}$ of a general position homotopy $H$ between $f$ and $f_0$. (For details and definition of the signs of the simplices see \cite{Hudson1969}, \S12, \cite{Vrabec1977}, p. 145, \cite{Skopenkov2006}, \S2, \cite{Skopenkov2010}.) It is for being well-defined that we need $\Zz_2$-coefficients when $n$ is even. *Clearly, $W(f_0)=0$. The definition of $W$ depends on the choice of $f_0$, but we write $W$ not $W_{f_0}$ for brevity. *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 $\sigma:\Rr^{2n}\to\Rr^{2n}$ 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 $\Rr^{2n}$ forces a change of the orientation of $f\cap F$). *The above definition makes sense for each $n$, not only for $n\ge3$. *Clearly, $W(\Hud_n(a))$ is $a$ or $a\mod2$ for $n\ge2$ for the [[Embeddings just below the stable range#Hudson tori 1|Hudson tori]]. *$W(f\#g)=W(f)$ for each embeddings $f:N\to\Rr^{2n}$ and $g:S^n\to\Rr^{2n}$. {{endthm}} == A generalization to highly-connected manifolds == ; Examples are the above Hudson tori $\Hud_{m,n,p}(a), \Hud'_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$. See also \cite{Milgram&Rees1971}. === Classification === ; {{beginthm|Theorem}}\label{hico} Let $N$ be a closed orientable homologically $k$-connected $n$-manifold, $k\ge0$. Then 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. E.g. by Theorem \ref{hico} we obtain that 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$$n>1$, then there is just one isotopy class of embedding $N \to \Rr^m$$N \to \Rr^m$ if $m \geq 2n + 1$$m \geq 2n + 1$. In this page we summarise the situation for $m = 2n\ge8$$m = 2n\ge8$, and give references to the case $m=2n-1\ge9$$m=2n-1\ge9$.

For notation and conventions see high codimension embeddings.

2 Classification

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 [Haefliger&Hirsch1963], [Bausum1975], [Vrabec1977], cf. [Hudson1969]

The Whitney invariant is defined below.

The classification of smooth embeddings of 3-manifolds in 6-space is more complicated. For analogous classification of $E^{2n-k}(N)$$E^{2n-k}(N)$ see Embeddings of highly-connected manifolds. Some estimations of $E^{2n-k-1}(N)$$E^{2n-k-1}(N)$ for a closed $k$$k$-connected $n$$n$-manifold $N$$N$ and $n\ge2k+5$$n\ge2k+5$ (including $k=0$$k=0$) are presented in [Skopenkov2010]. See also Embeddings of 4-manifolds in 7-space.

3 Examples

Together with the Haefliger knotted sphere $S^{2l-1}\to S^{3l}$$S^{2l-1}\to S^{3l}$, Hudson's examples 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 the one below).

For $m\ge n+2$$m\ge n+2$ define the $standard$$standard$ $embedding$$embedding$ $S^p\times S^{n-p}\to S^m$$S^p\times S^{n-p}\to S^m$ as the composition $S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}\to\Rr^m\to S^m$$S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}\to\Rr^m\to S^m$ of standard embeddings.

3.1 Hudson tori 1

In this subsection, we recall for $a\in\Zz$$a\in\Zz$ and $n\ge2$$n\ge2$. Hudson's construction of embedings

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

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}$$\partial D^2\subset\partial D^{n+1}$. Fix a point $x\in S^{n-1}$$x\in S^{n-1}$. The Hudson torus $\Hud_n(1)$$\Hud_n(1)$ is the embedded connected sum of

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

(Unlike the connected sum mentioned in embedded conntected sum this is a `linked' connected sum, i.e. connected sum of two embeddings whose images are not contained in disjoint cubes.)

For $a\in\Zz$$a\in\Zz$ instead of an embedded $n$$n$-sphere $2\partial D^{n+1}\times x$$2\partial D^{n+1}\times x$ we can take $a$$a$ copies $(1+\frac1k)\partial D^{n+1}\times x$$(1+\frac1k)\partial D^{n+1}\times x$ ($k=1,\dots,a$$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 x$$\partial D^{n+1}\times x$. Then we join these spheres by tubes so that the homotopy class of the resulting embedding

$\displaystyle S^n\to S^{2n}-D^{n+1}\times S^{n-1}\simeq S^{2n}-S^{n-1}\simeq S^n \quad\text{will be}\quad a\in\pi_n(S^n)\cong\Zz.$ $$\displaystyle S^n\to S^{2n}-D^{n+1}\times S^{n-1}\simeq S^{2n}-S^{n-1}\simeq S^n \quad\text{will be}\quad a\in\pi_n(S^n)\cong\Zz.$$

Let $\Hud_n(a)$$\Hud_n(a)$ be the connected sum of this embedding with the above standard embedding $\partial D^2\times S^{n-1}\subset\Rr^{2n}$$\partial D^2\times S^{n-1}\subset\Rr^{2n}$.

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

Proposition 3.1. 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$.

.

In particular, $\Hud_n(1)$$\Hud_n(1)$ is not isotopic to $\Hud_n(0)$$\Hud_n(0)$ for each $n$$n$ (this was the original motivation for Hudson).

Proposition 3.1 follows by Remark 5.e and, for $n$$n$ even, by Theorem 4, both results from classification just below the stable range.

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], \S6) and to prove the analogue of Proposition 3.1 for $n=2$$n=2$.

3.2 Hudson tori 2

In this subsection we 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}$$1\times S^{n-1}$ and the standard embedding $\varphi:\{-1\}\times S^{n-1}\to\partial D^n\subset D^n$$\varphi:\{-1\}\times S^{n-1}\to\partial D^n\subset D^n$ on the other component. This map gives an $embedding$$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].) Each 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$. Extend this embedding $S^0\to D^n\times x$$S^0\to D^n\times x$ for each $x\in S^{n-1}$$x\in S^{n-1}$ to an embedding $S^1\to D^{n+1}\times x$$S^1\to D^{n+1}\times x$. (See Figure 2.3 of [Skopenkov2006].) Thus we obtain the Hudson torus

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

The embedding $\Hud_n'(a)$$\Hud_n'(a)$ is obtained in the same way starting from a map $\varphi$$\varphi$ of degree $a$$a$.

The same proposition as above holds with $\Hud_n$$\Hud_n$ replaced to $\Hud_n'$$\Hud_n'$.

3.3 Remarks

We have $\Hud_n(a)$$\Hud_n(a)$ is PL isotopic to $\Hud_n'(a)$$\Hud_n'(a)$ [Skopenkov2006a]. It would be interesting to prove the smooth analogue of this result.

For $n=1$$n=1$ these construction give what we call the $left$$left$ Hudson torus. The $right$$right$ Hudson torus is constructed analogously and is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of $S^1\times S^1$$S^1\times S^1$.

Analogously one constructs the Hudson torus $\Hud_{4,2}(a):S^2\times S^2\to\Rr^7$$\Hud_{4,2}(a):S^2\times S^2\to\Rr^7$ for $a\in\Zz$$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})$ and $\Hud_{m,n,p}'(a):S^p\times S^{n-p}\to\Rr^m$$\Hud_{m,n,p}'(a):S^p\times S^{n-p}\to\Rr^m$ for $a\in\pi_{n-p}(S^{m-n-1})$$a\in\pi_{n-p}(S^{m-n-1})$.

3.4 An action of the first homology group on embeddings

In this subsection, for $a\in H_1(N)$$a\in H_1(N)$ and for orientable $N$$N$ with $n\ge3$$n\ge3$, we construct an embedding $f_a:N\to\Rr^{2n}$$f_a:N\to\Rr^{2n}$ from an embedding $f_0:N\to\Rr^{2n}$$f_0:N\to\Rr^{2n}$.

For $a \in H_1(N)$$a \in H_1(N)$, represent $a$$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}$$S^1\times S^{n-1}$. It remains to make an embedded surgery of $S^1\times*\subset S^1\times S^{n-1}$$S^1\times*\subset S^1\times S^{n-1}$ to obtain an $n$$n$-sphere $\Sigma\subset C_{f_0}$$\Sigma\subset C_{f_0}$, and then we set $f_a:=f_0\# \Sigma$$f_a:=f_0\# \Sigma$.

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*$$S^1\times*$ along this vector field to a smooth map $\overline a:D^2\to S^{2n}$$\overline a:D^2\to S^{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

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

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

This construction generalizes the construction of $\Hud_n(a)$$\Hud_n(a)$ (from $\Hud_n(0)$$\Hud_n(0)$).

Clearly, $W(f_a)$$W(f_a)$ is $a$$a$ or $a\mod2$$a\mod2$. Thus unless $n=3$$n=3$ and CAT=DIFF

• all isotopy classes of embedings $N\to\Rr^{2n}$$N\to\Rr^{2n}$ can be obtained (from a certain given embedding $f_0$$f_0$) by the above construction;
• the above construction defines an action $H_1(N;\Zz_{(n-1)})\to E^{2n}(N)$$H_1(N;\Zz_{(n-1)})\to E^{2n}(N)$.

4 The Whitney invariant (for either n odd or N orientable)

Fix orientations on $\Rr^{2n}$$\Rr^{2n}$ and, if $N$$N$ is even, on $N$$N$. Fix an embedding $f_0:N\to\Rr^{2n}$$f_0:N\to\Rr^{2n}$. For an embedding $f:N\to\Rr^{2n}$$f:N\to\Rr^{2n}$ the restrictions of $f$$f$ and $f_0$$f_0$ to $N_0$$N_0$ are regular homotopic [Hirsch1959]. Since $N_0$$N_0$ has an $(n-1)$$(n-1)$-dimensional spine, it follows that 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$$N_0$. Take a general position homotopy $F:B^n\times I\to\Rr^{2n}$$F:B^n\times I\to\Rr^{2n}$ relative to $\partial B^n$$\partial B^n$ between the restrictions of $f$$f$ and $g$$g$ to $B^n$$B^n$. Then $f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$$f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$ (i.e. `the intersection of this homotopy with $f(N-B^n)$$f(N-B^n)$') is a 1-manifold (possibly non-compact) without boundary. Define $W(f)$$W(f)$ to be the homology class of the closure of this 1-manifold:

$\displaystyle W(f):=[Cl(f\cap F)]\in H_1(N_0,\partial N_0;\Zz_{(n)})\cong H_1(N;\Zz_{(n)}).$ $$\displaystyle W(f):=[Cl(f\cap F)]\in H_1(N_0,\partial N_0;\Zz_{(n)})\cong H_1(N;\Zz_{(n)}).$$

The orientation on $f\cap F$$f\cap F$ is defined for $N$$N$ orientable as follows. (This orientation is defined for each $n$$n$ but used only for odd $n$$n$.) 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)>2\cdot2n$$n+2(n+1)>2\cdot2n$, by general position there is a unique point $y\in B^n\times I$$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^{2n}$$\Rr^{2n}$. If this base is positive, then call the initial vector of $f\cap F$$f\cap F$ positive. Since a change of the orientation on $f\cap F$$f\cap F$ forces a change of the orientation of the latter base of $\Rr^{2n}$$\Rr^{2n}$, it follows that this condition indeed defines an orientation on $f\cap F$$f\cap F$.

Remark 4.1.

• The Whitney invariant is well-defined, i.e. independent of the choice of $F$$F$ and of the isotopy making $f=f_0$$f=f_0$ outside $B^n$$B^n$. This is so because the above definition is clearly equivalent to the following: $W(f)$$W(f)$ is the homology class of the algebraic sum of the top-dimensional simplices of the self-intersection set $\Sigma(H):=Cl\{x\in N\times I\ |\ \#H^{-1}Hx>1\}$$\Sigma(H):=Cl\{x\in N\times I\ |\ \#H^{-1}Hx>1\}$ of a general position homotopy $H$$H$ between $f$$f$ and $f_0$$f_0$. (For details and definition of the signs of the simplices see [Hudson1969], \S12, [Vrabec1977], p. 145, [Skopenkov2006], \S2, [Skopenkov2010].) It is for being well-defined that we need $\Zz_2$$\Zz_2$-coefficients when $n$$n$ is even.
• Clearly, $W(f_0)=0$$W(f_0)=0$. The definition of $W$$W$ depends on the choice of $f_0$$f_0$, but we write $W$$W$ not $W_{f_0}$$W_{f_0}$ for brevity.
• Since a change of the orientation on $N$$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$$N$. For the reflection $\sigma:\Rr^{2n}\to\Rr^{2n}$$\sigma:\Rr^{2n}\to\Rr^{2n}$ 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$$N_0$ and because a change of the orientation of $\Rr^{2n}$$\Rr^{2n}$ forces a change of the orientation of $f\cap F$$f\cap F$).
• The above definition makes sense for each $n$$n$, not only for $n\ge3$$n\ge3$.
• Clearly, $W(\Hud_n(a))$$W(\Hud_n(a))$ is $a$$a$ or $a\mod2$$a\mod2$ for $n\ge2$$n\ge2$ for the Hudson tori.
• $W(f\#g)=W(f)$$W(f\#g)=W(f)$ for each embeddings $f:N\to\Rr^{2n}$$f:N\to\Rr^{2n}$ and $g:S^n\to\Rr^{2n}$$g:S^n\to\Rr^{2n}$.

5 A generalization to highly-connected manifolds

Examples are the above 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$. See also [Milgram&Rees1971].

5.1 Classification

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$$homotopically$ $k$$k$-connected manifolds. The proof works for $homologically$$homologically$ $k$$k$-connected manifolds.

E.g. by Theorem 5.1 we obtain that 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; the generator is the Hudson torus.

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$.

Analogously to Theorem 5.1 it may be proved that

• if $N$$N$ is a closed connected non-orientable $n$$n$-manifold, then

$\displaystyle E^{2n}(N)=\begin{cases} H_1(N,\Zz_2)& n\text{ odd, }\\ \Zz\oplus\Zz_2^{s-1}&n\text{ even and }H_1(N,\Zz_2)\cong\Zz_2^s\end{cases},$ $$\displaystyle E^{2n}(N)=\begin{cases} H_1(N,\Zz_2)& n\text{ odd, }\\ \Zz\oplus\Zz_2^{s-1}&n\text{ even and }H_1(N,\Zz_2)\cong\Zz_2^s\end{cases},$$

provided $n\ge3$$n\ge3$ or $n\ge4$$n\ge4$ in the PL or DIFF categories, respectively.

This is proved in [Haefliger1962b], [Weber1967], [Bausum1975], [Vrabec1977] (a minor miscalculation being corrected only in [Vrabec1977]).

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 and $n\le2k+3$$n\le2k+3$ a classification is much harder: for 40 years the $only$$only$ known concrete complete classification results were for spheres. The following result was obtained using the Kreck modification of surgery theory.

Theorem 5.2. [Skopenkov2006] Let $N$$N$ be a closed homologically $(2k-2)$$(2k-2)$-connected $(4k-1)$$(4k-1)$-manifold. Then the Whitney invariant $W:E^{6k}_D(N)\to H_{2k-1}(N)$$W:E^{6k}_D(N)\to H_{2k-1}(N)$ is surjective and for each $u\in H_{2k-1}(N)$$u\in H_{2k-1}(N)$ there is a 1--1 correspondence $\eta_u:W^{-1}u\to\Zz_{d(u)}$$\eta_u:W^{-1}u\to\Zz_{d(u)}$, 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 we obtain that

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

5.2 The Whitney invariant

6 References

• [Bausum1975] D. R. Bausum, Embeddings and immersions of manifolds in Euclidean space, Trans. Amer. Math. Soc. 213 (1975), 263–303. MR0474330 (57 #13976) Zbl 0323.57017
• [Boechat&Haefliger1970] J. Boéchat and A. Haefliger, Plongements différentiables des variétés orientées de dimension $4$$4$ dans $\Rr^7$$\Rr^7$, (French) Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York (1970), 156–166. MR0270384 (42 #5273) Zbl 0199.27003
• [Boechat1971] J. Boéchat, Plongements de variétées différentiables orientées de dimension $4k$$4k$ dans $\Rr^{6k+1}$$\Rr^{6k+1}$, Comment. Math. Helv. 46 (1971), 141–161. MR0295373 (45 #4439) Zbl 0218.57016
• [Haefliger&Hirsch1963] A. Haefliger and M. W. Hirsch, On the existence and classification of differentiable embeddings, Topology 2 (1963), 129–135. see also MR0149494 (26 #6981) Zbl 0113.38607
• [Haefliger1962b] A. Haefliger, Plongements de variétés dans le domain stable, Séminaire Bourbaki, 245 (1962).
• [Hirsch1959] M. W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR0119214 (22 #9980) Zbl 0118.18603
• [Hudson1963] J. F. P. Hudson, Knotted tori, Topology 2 (1963), 11–22. MR0146829 (26 #4349) Zbl 0151.32503
• [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
• [Milgram&Rees1971] R. Milgram and E. Rees, On the normal bundle to an embedding., Topology 10 (1971), 299-308. MR0290391 (44 #7572) Zbl 0207.22302
• [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
• [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
• [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into $\Rr^{2n-k-1}$$\Rr^{2n-k-1}$, Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
• [Takase2006] M. Takase, Homology 3-spheres in codimension three, Internat. J. Math. 17 (2006), no.8, 869–885.

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

• [Vrabec1977] J. Vrabec, Knotting a $k$$k$-connected closed $\PL$$\PL$ $m$$m$-manifold in $E^{2m-k}$$E^{2m-k}$, Trans. Amer. Math. Soc. 233 (1977), 137–165. MR0645405 (58 #31097) Zbl 386.57013
• [Weber1967] C. Weber, Plongements de polyedres dans le domaine metastable, Comment. Math. Helv. 42 (1967), 1-27. MR0238330 (38 #6606) Zbl 0152.22402

This page has not been refereed. The information given here might be incomplete or provisional.

\le p\le q-2$. It is in fact a group isomorphism; the generator is the Hudson torus. The PL case of Theorem \ref{hico} gives nothing but [[High codimension embeddings: classification#Unknotting theorems|the Unknotting Spheres Theorem]] for $k+3\le n\le2k+1$. Analogously to Theorem \ref{hico} it may be proved that * if $N$ is a closed connected non-orientable $n$-manifold, then $$E^{2n}(N)=\begin{cases} H_1(N,\Zz_2)& n\text{ odd, }\ \Zz\oplus\Zz_2^{s-1}&n\text{ even and }H_1(N,\Zz_2)\cong\Zz_2^s\end{cases},$$ provided $n\ge3$ or $n\ge4$ in the PL or DIFF categories, respectively. This is proved in \cite{Haefliger1962b}, \cite{Weber1967}, \cite{Bausum1975}, \cite{Vrabec1977} (a minor miscalculation being corrected only in \cite{Vrabec1977}). 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 and $n\le2k+3$ a classification is much harder: for 40 years the $only$ known concrete complete classification results were for spheres. The following result was obtained using the Kreck modification of surgery theory. {{beginthm|Theorem}}\label{hicod} \cite{Skopenkov2006} Let $N$ be a closed homologically $(2k-2)$-connected $(4k-1)$-manifold. Then the Whitney invariant $W:E^{6k}_D(N)\to H_{2k-1}(N)$ is surjective and for each $u\in H_{2k-1}(N)$ there is a 1--1 correspondence $\eta_u:W^{-1}u\to\Zz_{d(u)}$, 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} we obtain that * the Whitney invariant $W:E^{6k}(S^{2k-1}\times S^{2k})\to\Zz$ is surjective and for each $u\in\Zz$ there is a 1--1 correspondence $W^{-1}u\to\Zz_u$. === The Whitney invariant === ; == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]] {{Stub}}N is a connected manifold of dimension $n>1$$n>1$, then there is just one isotopy class of embedding $N \to \Rr^m$$N \to \Rr^m$ if $m \geq 2n + 1$$m \geq 2n + 1$. In this page we summarise the situation for $m = 2n\ge8$$m = 2n\ge8$, and give references to the case $m=2n-1\ge9$$m=2n-1\ge9$.

For notation and conventions see high codimension embeddings.

2 Classification

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 [Haefliger&Hirsch1963], [Bausum1975], [Vrabec1977], cf. [Hudson1969]

The Whitney invariant is defined below.

The classification of smooth embeddings of 3-manifolds in 6-space is more complicated. For analogous classification of $E^{2n-k}(N)$$E^{2n-k}(N)$ see Embeddings of highly-connected manifolds. Some estimations of $E^{2n-k-1}(N)$$E^{2n-k-1}(N)$ for a closed $k$$k$-connected $n$$n$-manifold $N$$N$ and $n\ge2k+5$$n\ge2k+5$ (including $k=0$$k=0$) are presented in [Skopenkov2010]. See also Embeddings of 4-manifolds in 7-space.

3 Examples

Together with the Haefliger knotted sphere $S^{2l-1}\to S^{3l}$$S^{2l-1}\to S^{3l}$, Hudson's examples 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 the one below).

For $m\ge n+2$$m\ge n+2$ define the $standard$$standard$ $embedding$$embedding$ $S^p\times S^{n-p}\to S^m$$S^p\times S^{n-p}\to S^m$ as the composition $S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}\to\Rr^m\to S^m$$S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}\to\Rr^m\to S^m$ of standard embeddings.

3.1 Hudson tori 1

In this subsection, we recall for $a\in\Zz$$a\in\Zz$ and $n\ge2$$n\ge2$. Hudson's construction of embedings

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

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}$$\partial D^2\subset\partial D^{n+1}$. Fix a point $x\in S^{n-1}$$x\in S^{n-1}$. The Hudson torus $\Hud_n(1)$$\Hud_n(1)$ is the embedded connected sum of

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

(Unlike the connected sum mentioned in embedded conntected sum this is a `linked' connected sum, i.e. connected sum of two embeddings whose images are not contained in disjoint cubes.)

For $a\in\Zz$$a\in\Zz$ instead of an embedded $n$$n$-sphere $2\partial D^{n+1}\times x$$2\partial D^{n+1}\times x$ we can take $a$$a$ copies $(1+\frac1k)\partial D^{n+1}\times x$$(1+\frac1k)\partial D^{n+1}\times x$ ($k=1,\dots,a$$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 x$$\partial D^{n+1}\times x$. Then we join these spheres by tubes so that the homotopy class of the resulting embedding

$\displaystyle S^n\to S^{2n}-D^{n+1}\times S^{n-1}\simeq S^{2n}-S^{n-1}\simeq S^n \quad\text{will be}\quad a\in\pi_n(S^n)\cong\Zz.$ $$\displaystyle S^n\to S^{2n}-D^{n+1}\times S^{n-1}\simeq S^{2n}-S^{n-1}\simeq S^n \quad\text{will be}\quad a\in\pi_n(S^n)\cong\Zz.$$

Let $\Hud_n(a)$$\Hud_n(a)$ be the connected sum of this embedding with the above standard embedding $\partial D^2\times S^{n-1}\subset\Rr^{2n}$$\partial D^2\times S^{n-1}\subset\Rr^{2n}$.

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

Proposition 3.1. 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$.

.

In particular, $\Hud_n(1)$$\Hud_n(1)$ is not isotopic to $\Hud_n(0)$$\Hud_n(0)$ for each $n$$n$ (this was the original motivation for Hudson).

Proposition 3.1 follows by Remark 5.e and, for $n$$n$ even, by Theorem 4, both results from classification just below the stable range.

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], \S6) and to prove the analogue of Proposition 3.1 for $n=2$$n=2$.

3.2 Hudson tori 2

In this subsection we 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}$$1\times S^{n-1}$ and the standard embedding $\varphi:\{-1\}\times S^{n-1}\to\partial D^n\subset D^n$$\varphi:\{-1\}\times S^{n-1}\to\partial D^n\subset D^n$ on the other component. This map gives an $embedding$$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].) Each 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$. Extend this embedding $S^0\to D^n\times x$$S^0\to D^n\times x$ for each $x\in S^{n-1}$$x\in S^{n-1}$ to an embedding $S^1\to D^{n+1}\times x$$S^1\to D^{n+1}\times x$. (See Figure 2.3 of [Skopenkov2006].) Thus we obtain the Hudson torus

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

The embedding $\Hud_n'(a)$$\Hud_n'(a)$ is obtained in the same way starting from a map $\varphi$$\varphi$ of degree $a$$a$.

The same proposition as above holds with $\Hud_n$$\Hud_n$ replaced to $\Hud_n'$$\Hud_n'$.

3.3 Remarks

We have $\Hud_n(a)$$\Hud_n(a)$ is PL isotopic to $\Hud_n'(a)$$\Hud_n'(a)$ [Skopenkov2006a]. It would be interesting to prove the smooth analogue of this result.

For $n=1$$n=1$ these construction give what we call the $left$$left$ Hudson torus. The $right$$right$ Hudson torus is constructed analogously and is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of $S^1\times S^1$$S^1\times S^1$.

Analogously one constructs the Hudson torus $\Hud_{4,2}(a):S^2\times S^2\to\Rr^7$$\Hud_{4,2}(a):S^2\times S^2\to\Rr^7$ for $a\in\Zz$$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})$ and $\Hud_{m,n,p}'(a):S^p\times S^{n-p}\to\Rr^m$$\Hud_{m,n,p}'(a):S^p\times S^{n-p}\to\Rr^m$ for $a\in\pi_{n-p}(S^{m-n-1})$$a\in\pi_{n-p}(S^{m-n-1})$.

3.4 An action of the first homology group on embeddings

In this subsection, for $a\in H_1(N)$$a\in H_1(N)$ and for orientable $N$$N$ with $n\ge3$$n\ge3$, we construct an embedding $f_a:N\to\Rr^{2n}$$f_a:N\to\Rr^{2n}$ from an embedding $f_0:N\to\Rr^{2n}$$f_0:N\to\Rr^{2n}$.

For $a \in H_1(N)$$a \in H_1(N)$, represent $a$$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}$$S^1\times S^{n-1}$. It remains to make an embedded surgery of $S^1\times*\subset S^1\times S^{n-1}$$S^1\times*\subset S^1\times S^{n-1}$ to obtain an $n$$n$-sphere $\Sigma\subset C_{f_0}$$\Sigma\subset C_{f_0}$, and then we set $f_a:=f_0\# \Sigma$$f_a:=f_0\# \Sigma$.

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*$$S^1\times*$ along this vector field to a smooth map $\overline a:D^2\to S^{2n}$$\overline a:D^2\to S^{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

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

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

This construction generalizes the construction of $\Hud_n(a)$$\Hud_n(a)$ (from $\Hud_n(0)$$\Hud_n(0)$).

Clearly, $W(f_a)$$W(f_a)$ is $a$$a$ or $a\mod2$$a\mod2$. Thus unless $n=3$$n=3$ and CAT=DIFF

• all isotopy classes of embedings $N\to\Rr^{2n}$$N\to\Rr^{2n}$ can be obtained (from a certain given embedding $f_0$$f_0$) by the above construction;
• the above construction defines an action $H_1(N;\Zz_{(n-1)})\to E^{2n}(N)$$H_1(N;\Zz_{(n-1)})\to E^{2n}(N)$.

4 The Whitney invariant (for either n odd or N orientable)

Fix orientations on $\Rr^{2n}$$\Rr^{2n}$ and, if $N$$N$ is even, on $N$$N$. Fix an embedding $f_0:N\to\Rr^{2n}$$f_0:N\to\Rr^{2n}$. For an embedding $f:N\to\Rr^{2n}$$f:N\to\Rr^{2n}$ the restrictions of $f$$f$ and $f_0$$f_0$ to $N_0$$N_0$ are regular homotopic [Hirsch1959]. Since $N_0$$N_0$ has an $(n-1)$$(n-1)$-dimensional spine, it follows that 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$$N_0$. Take a general position homotopy $F:B^n\times I\to\Rr^{2n}$$F:B^n\times I\to\Rr^{2n}$ relative to $\partial B^n$$\partial B^n$ between the restrictions of $f$$f$ and $g$$g$ to $B^n$$B^n$. Then $f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$$f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$ (i.e. `the intersection of this homotopy with $f(N-B^n)$$f(N-B^n)$') is a 1-manifold (possibly non-compact) without boundary. Define $W(f)$$W(f)$ to be the homology class of the closure of this 1-manifold:

$\displaystyle W(f):=[Cl(f\cap F)]\in H_1(N_0,\partial N_0;\Zz_{(n)})\cong H_1(N;\Zz_{(n)}).$ $$\displaystyle W(f):=[Cl(f\cap F)]\in H_1(N_0,\partial N_0;\Zz_{(n)})\cong H_1(N;\Zz_{(n)}).$$

The orientation on $f\cap F$$f\cap F$ is defined for $N$$N$ orientable as follows. (This orientation is defined for each $n$$n$ but used only for odd $n$$n$.) 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)>2\cdot2n$$n+2(n+1)>2\cdot2n$, by general position there is a unique point $y\in B^n\times I$$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^{2n}$$\Rr^{2n}$. If this base is positive, then call the initial vector of $f\cap F$$f\cap F$ positive. Since a change of the orientation on $f\cap F$$f\cap F$ forces a change of the orientation of the latter base of $\Rr^{2n}$$\Rr^{2n}$, it follows that this condition indeed defines an orientation on $f\cap F$$f\cap F$.

Remark 4.1.

• The Whitney invariant is well-defined, i.e. independent of the choice of $F$$F$ and of the isotopy making $f=f_0$$f=f_0$ outside $B^n$$B^n$. This is so because the above definition is clearly equivalent to the following: $W(f)$$W(f)$ is the homology class of the algebraic sum of the top-dimensional simplices of the self-intersection set $\Sigma(H):=Cl\{x\in N\times I\ |\ \#H^{-1}Hx>1\}$$\Sigma(H):=Cl\{x\in N\times I\ |\ \#H^{-1}Hx>1\}$ of a general position homotopy $H$$H$ between $f$$f$ and $f_0$$f_0$. (For details and definition of the signs of the simplices see [Hudson1969], \S12, [Vrabec1977], p. 145, [Skopenkov2006], \S2, [Skopenkov2010].) It is for being well-defined that we need $\Zz_2$$\Zz_2$-coefficients when $n$$n$ is even.
• Clearly, $W(f_0)=0$$W(f_0)=0$. The definition of $W$$W$ depends on the choice of $f_0$$f_0$, but we write $W$$W$ not $W_{f_0}$$W_{f_0}$ for brevity.
• Since a change of the orientation on $N$$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$$N$. For the reflection $\sigma:\Rr^{2n}\to\Rr^{2n}$$\sigma:\Rr^{2n}\to\Rr^{2n}$ 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$$N_0$ and because a change of the orientation of $\Rr^{2n}$$\Rr^{2n}$ forces a change of the orientation of $f\cap F$$f\cap F$).
• The above definition makes sense for each $n$$n$, not only for $n\ge3$$n\ge3$.
• Clearly, $W(\Hud_n(a))$$W(\Hud_n(a))$ is $a$$a$ or $a\mod2$$a\mod2$ for $n\ge2$$n\ge2$ for the Hudson tori.
• $W(f\#g)=W(f)$$W(f\#g)=W(f)$ for each embeddings $f:N\to\Rr^{2n}$$f:N\to\Rr^{2n}$ and $g:S^n\to\Rr^{2n}$$g:S^n\to\Rr^{2n}$.

5 A generalization to highly-connected manifolds

Examples are the above 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$. See also [Milgram&Rees1971].

5.1 Classification

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$$homotopically$ $k$$k$-connected manifolds. The proof works for $homologically$$homologically$ $k$$k$-connected manifolds.

E.g. by Theorem 5.1 we obtain that 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; the generator is the Hudson torus.

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$.

Analogously to Theorem 5.1 it may be proved that

• if $N$$N$ is a closed connected non-orientable $n$$n$-manifold, then

$\displaystyle E^{2n}(N)=\begin{cases} H_1(N,\Zz_2)& n\text{ odd, }\\ \Zz\oplus\Zz_2^{s-1}&n\text{ even and }H_1(N,\Zz_2)\cong\Zz_2^s\end{cases},$ $$\displaystyle E^{2n}(N)=\begin{cases} H_1(N,\Zz_2)& n\text{ odd, }\\ \Zz\oplus\Zz_2^{s-1}&n\text{ even and }H_1(N,\Zz_2)\cong\Zz_2^s\end{cases},$$

provided $n\ge3$$n\ge3$ or $n\ge4$$n\ge4$ in the PL or DIFF categories, respectively.

This is proved in [Haefliger1962b], [Weber1967], [Bausum1975], [Vrabec1977] (a minor miscalculation being corrected only in [Vrabec1977]).

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 and $n\le2k+3$$n\le2k+3$ a classification is much harder: for 40 years the $only$$only$ known concrete complete classification results were for spheres. The following result was obtained using the Kreck modification of surgery theory.

Theorem 5.2. [Skopenkov2006] Let $N$$N$ be a closed homologically $(2k-2)$$(2k-2)$-connected $(4k-1)$$(4k-1)$-manifold. Then the Whitney invariant $W:E^{6k}_D(N)\to H_{2k-1}(N)$$W:E^{6k}_D(N)\to H_{2k-1}(N)$ is surjective and for each $u\in H_{2k-1}(N)$$u\in H_{2k-1}(N)$ there is a 1--1 correspondence $\eta_u:W^{-1}u\to\Zz_{d(u)}$$\eta_u:W^{-1}u\to\Zz_{d(u)}$, 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 we obtain that

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

6 References

• [Bausum1975] D. R. Bausum, Embeddings and immersions of manifolds in Euclidean space, Trans. Amer. Math. Soc. 213 (1975), 263–303. MR0474330 (57 #13976) Zbl 0323.57017
• [Boechat&Haefliger1970] J. Boéchat and A. Haefliger, Plongements différentiables des variétés orientées de dimension $4$$4$ dans $\Rr^7$$\Rr^7$, (French) Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York (1970), 156–166. MR0270384 (42 #5273) Zbl 0199.27003
• [Boechat1971] J. Boéchat, Plongements de variétées différentiables orientées de dimension $4k$$4k$ dans $\Rr^{6k+1}$$\Rr^{6k+1}$, Comment. Math. Helv. 46 (1971), 141–161. MR0295373 (45 #4439) Zbl 0218.57016
• [Haefliger&Hirsch1963] A. Haefliger and M. W. Hirsch, On the existence and classification of differentiable embeddings, Topology 2 (1963), 129–135. see also MR0149494 (26 #6981) Zbl 0113.38607
• [Haefliger1962b] A. Haefliger, Plongements de variétés dans le domain stable, Séminaire Bourbaki, 245 (1962).
• [Hirsch1959] M. W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR0119214 (22 #9980) Zbl 0118.18603
• [Hudson1963] J. F. P. Hudson, Knotted tori, Topology 2 (1963), 11–22. MR0146829 (26 #4349) Zbl 0151.32503
• [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
• [Milgram&Rees1971] R. Milgram and E. Rees, On the normal bundle to an embedding., Topology 10 (1971), 299-308. MR0290391 (44 #7572) Zbl 0207.22302
• [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
• [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
• [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into $\Rr^{2n-k-1}$$\Rr^{2n-k-1}$, Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
• [Takase2006] M. Takase, Homology 3-spheres in codimension three, Internat. J. Math. 17 (2006), no.8, 869–885.

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

• [Vrabec1977] J. Vrabec, Knotting a $k$$k$-connected closed $\PL$$\PL$ $m$$m$-manifold in $E^{2m-k}$$E^{2m-k}$, Trans. Amer. Math. Soc. 233 (1977), 137–165. MR0645405 (58 #31097) Zbl 386.57013
• [Weber1967] C. Weber, Plongements de polyedres dans le domaine metastable, Comment. Math. Helv. 42 (1967), 1-27. MR0238330 (38 #6606) Zbl 0152.22402

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