Embeddings just below the stable range: classification

Revision as of 13:06, 27 April 2016 (view source) Askopenkov (Talk | contribs) (→Examples) ← Older edit		Revision as of 13:07, 27 April 2016 (view source) Askopenkov (Talk | contribs) (→Action by linked embedded connected sum) Newer edit →
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	Then we define $f_a$ to be the (linked) embedded connected sum of $f_0$ and $g$ (along certain arc joining their images).		Then we define $f_a$ to be the (linked) embedded connected sum of $f_0$ and $g$ (along certain arc joining their images).
−	Take a vector field on $S^1\times*$ normal to $S^1\times S^{n-1}$. Extend $S^1\times*$ along this vector field to a smooth map $\overline a:D^2\to S^{2n}$. Since $2n>4$ 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	+	Take a vector field on $S^1\times*$ normal to $S^1\times S^{n-1}$.
−	$$\widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad	+	Extend $S^1\times*$ along this vector field to a map $\overline a:D^2\to\Rr^{2n}$.
−	\widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}.$$	+	Since $2n>4$ and $n+2<2n$, by general position we may assume that $\overline a$ is an embedding and $\overline a(Int D^2)$ misses $f_0(N)\cup S^1\times S^{n-1}$.
		+	Since $n-1>1$, we have $\pi_1(V_{2n-2,n-1})=0$.
		+	Hence the standard framing of $S^1\times*$ in $S^1\times S^{n-1}$ extends to an $(n-1)$-framing on $\overline a(D^2)$ in $\Rr^{2n}$.
		+	Thus $\overline a$ extends to an embedding
		+	$$
		+	\widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad
		+	\widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}.
		+	$$
	Define embedding $g:S^n\to C_{f_0}$ by setting		Define embedding $g:S^n\to C_{f_0}$ by setting
−	$$g(S^n):\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1})	+	$$
		+	g(S^n):\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1})
	\bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})}		\bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})}
−	\widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n$$	+	\widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n
		+	$$
	with natural orientation.		with natural orientation.

---

Revision as of 13:07, 27 April 2016

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Contents 1 Introduction 2 Classification 3 Examples 3.1 Hudson tori 3.2 Action by linked embedded connected sum 4 The Whitney invariant (for either n even or N orientable) 5 A generalization to highly-connected manifolds 5.1 Classification 5.2 The Whitney invariant 6 An orientation on the self-intersection set 7 References

1 Introduction

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

Recall the unknotting theorem that if ${{Stub}} == Introduction == <wikitex>; This page is intended not only for specialists in embeddings, but also for mathematician from other areas who want to apply or to learn theory of embeddings. Recall the [[High codimension embeddings#Unknotting theorems|unknotting theorem]] that if $N$ is a connected manifold of dimension $n>1$, and $m \ge2n+1$, then every two embeddings $N \to\Rr^m$ are isotopic. In this page we summarise the situation for $m=2n\ge6$ and some more general situations. See [[High_codimension_embeddings|general introduction on embeddings, notation and conventions]]. </wikitex> == Classification == <wikitex>; {{beginthm|Theorem}}\label{th4} Let $N$ be a closed connected $n$-manifold. The [[Embeddings just below the stable range: classification#The Whitney invariant (for either n even or N orientable)|Whitney invariant]] $$W:E^{2n}(N)\to\left\{\begin{array}{cc} H_1(N;\Zz_{(n-1)})& \mbox{either $n$ is odd or $N$ is orientable}\ \Zz\oplus\Zz_2^{s-1}&\mbox{$n$ is even, $N$ is non-orientable and } H_1(N,\Zz_2)\cong\Zz_2^s\end{array}\right.$$ is bijective if either $n\ge4$ or $n=3$ and CAT=PL. {{endthm}} This is proved in \cite{Haefliger1962b}, \cite{Haefliger&Hirsch1963}, \cite{Weber1967}, \cite{Bausum1975}, \cite{Vrabec1977} (a minor miscalculation for the non-orientable case being corrected only in \cite{Vrabec1977}). The classification of ''smooth'' [[Embeddings of 3-manifolds in 6-space|embeddings of 3-manifolds in $\Rr^6$]] is more complicated. For embeddings of $n$-manifolds in $\Rr^{2n-1}$ see [[Embeddings_of_4-manifolds_in_7-space|the case of 4-manifolds]], \cite{Yasui1984}, for $n\ge5$ and \cite{Saeki1999}, \cite{Skopenkov2010}, \cite{Tonkonog2010} for non-closed manifolds. Theorem \ref{th4} is generalized to [[Embeddings just below the stable range: classification#A generalization to highly-connected manifolds|a description]] of $E^{2n-k}(N)$ for closed $k$-connected $n$-manifolds $N$. </wikitex> == Examples == <wikitex>; Together with [[Knots,_i.e._embeddings_of_spheres#The_Haefliger_trefoil_knot|the Haefliger knotted sphere]], examples of Hudson tori presented below were the first examples of embeddings in codimension greater than 2 not isotopic to the standard embedding. (Hudson's construction \cite{Hudson1963} was not as explicit as those below.) For $m\ge n+2$ we define the ''standard embedding'' $S^p\times S^{n-p}\to\Rr^m$ as the composition $S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}=\Rr^{n+2}\to\Rr^m$ of standard embeddings. Let $*:=(-1,0,\ldots,0)\in S^{n-1}$. </wikitex> === Hudson tori=== <wikitex>; In this subsection we construct, for $a\in\Zz$ and $n\ge2$, an embedding $$\Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.$$ The reader might first consider the case $n=2$. {{beginthm|Definition}}\label{dh11} (This construction, as opposed to Definition \ref{dh1n}, works for $n=1$.) Take the standard embeddings D^{n+1}\times S^{n-1}\subset\Rr^{2n}$ (where $ means homothety with coefficient 2) and $\partial D^2\subset\partial D^{n+1}$. Take embedded sphere and embedded torus $\partial D^{n+1}\times *\subset 2D^{n+1}\times S^{n-1}\subset\Rr^{2n}\quad\text{and}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$ Join them by an arc whose interior misses the two embedded manifolds. ''The Hudson torus'' $\Hud_n(1)$ is the embedded connected sum of the two embeddings along this arc, compatible with the orientation. (Unlike the [[High_codimension_embeddings#Embedded connected sum|unlinked embedded connected sum]] this is a ''linked'' embedded connected sum, i.e. connected sum of two embeddings whose images are ''not'' contained in disjoint cubes.) {{endthm}} {{beginthm|Definition}}\label{dh1n} For $a\in\Zz$ instead of \partial D^{n+1}\times*$ we take $|a|$ copies $(1+\frac1k)\partial D^{n+1}\times*$ ($k=1,\dots,|a|$) of $n$-sphere outside $D^{n+1}\times S^{n-1}$ `parallel' to $\partial D^{n+1}\times *$, with standard orientation for $a>0$ or the opposite orientation for $a<0$. Then we make embedded connected sum by tubes joining each $k$-th copy to $(k+1)$-th copy. We obtain an embedding $g:S^n\to\Rr^{2n}$. Let $\Hud_n(a)$ be the linked embedded connected sum of $g$ with the embedding $\partial D^2\times S^{n-1}\subset\Rr^{2n}$ from Definition \ref{dh11}. {{endthm}} Clearly, $\Hud_n(0)$ is isotopic to the standard embedding. The original motivation for Hudson was that $\Hud_n(1)$ is not isotopic to $\Hud_n(0)$ for each $n$ (this is a particular case of Proposition \ref{pr3} below). One guesses that $\Hud_n(a)$ is not isotopic to $\Hud_n(a')$ for $a\ne a'$. And that a $\Zz$-valued invariant exists and is `realized' by the homotopy class of the map <!--the abbreivation of embedding $g$ from Definition \ref{dh1n}--> $$S^n\overset g\to S^{2n}-D^{n+1}\times S^{n-1}\sim S^{2n}-S^{n-1}\sim S^n \quad\text{which is}\quad a\in\pi_n(S^n)\cong\Zz.$$ However, this is only true for $n$ odd. {{beginthm|Proposition}}\label{pr3} For $n\ge3$ odd $\Hud_n(a)$ is isotopic to $\Hud_n(a')$ if and only if $a=a'$. For $n\ge4$ even $\Hud_n(a)$ is isotopic to $\Hud_n(a')$ if and only if $a\equiv a'\mod2$. {{endthm}} Proposition \ref{pr3} follows by calculation of [[Embeddings_just_below_the_stable_range:_classification#The_Whitney_invariant_.28for_either_n_even_or_N_orientable.29|the Whitney invariant]] (Remark \ref{re5}.e below) and, for $n$ even, by Theorem \ref{th4}. Analogously, $\Hud_2(a)$ is not isotopic to $\Hud_2(a')$ if $a\not\equiv a'\mod2$. It would be interesting to know if the converse holds. E.g. is $\Hud_2(0)$ isotopic to $\Hud_2(2)$? It would also be interesting to find an explicit construction of an isotopy between $\Hud_{2k}(a)$ and $\Hud_{2k}(a+2)$ (cf. \cite{Vrabec1977}, \S6). {{beginthm|Definition}}\label{dh2n} Let us give, for $a\in\Zz$ and $n\ge2$, another construction of embeddings $$\Hud_n'(a):S^1\times S^{n-1}\to\Rr^{2n}.$$ Define a map $S^0\times S^{n-1}\to D^n$ to be the constant <script type="math/tex">N$ is a connected manifold of dimension $n>1$, and $m \ge2n+1$, then every two embeddings $N \to\Rr^m$ are isotopic. In this page we summarise the situation for $m=2n\ge6$ and some more general situations.

See general introduction on embeddings, notation and conventions.

2 Classification

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

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

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

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

3 Examples

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

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

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

3.1 Hudson tori

In this subsection we construct, for $a\in\Zz$ and $n\ge2$, an embedding

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

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

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

One guesses that $\Hud_n(a)$ is not isotopic to $\Hud_n(a')$ for $a\ne a'$. And that a $\Zz$-valued invariant exists and is `realized' by the homotopy class of the map

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

Proposition 3.3 follows by calculation of the Whitney invariant (Remark 4.1.e below) and, for $n$ even, by Theorem 2.1. Analogously, $\Hud_2(a)$ is not isotopic to $\Hud_2(a')$ if $a\not\equiv a'\mod2$. It would be interesting to know if the converse holds. E.g. is $\Hud_2(0)$ isotopic to $\Hud_2(2)$? It would also be interesting to find an explicit construction of an isotopy between $\Hud_{2k}(a)$ and $\Hud_{2k}(a+2)$ (cf. [Vrabec1977], \S6).

3.2 Action by linked embedded connected sum

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

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

Take a vector field on $S^1\times*$ normal to $S^1\times S^{n-1}$. Extend $S^1\times*$ along this vector field to a map $\overline a:D^2\to\Rr^{2n}$. Since $2n>4$ 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

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

with natural orientation.

By definition of the Whitney invariant, $W(f_a)$ 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 $H_1(N;\Zz_{(n-1)})\to E^{2n}(N)$.

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

Fix orientations on $\Rr^{2n}$ and, if $n$ is odd, 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 [Hirsch1959]. Since $N_0$ has an $(n-1)$-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 $f$ 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:

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 $f\cap F$.

5 A generalization to highly-connected manifolds

Let $N$ be a closed $k$-connected $n$-manifold. We present description of $E^{2n-k}(N)$ generalizing Theorem 2.1, and its generalization to $E^m(N)$ for $m\ge2n-2k+1$.

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

5.1 Classification

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

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

E.g. by Theorem 5.1 the Whitney invariant $W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{(q)}$ is bijective for $1\le p\le q-2$. It is in fact a group isomorphism; the generator is the Hudson torus.

Because of the existence of knots the analogues of Theorem 5.1 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 Bo\'echat-Haefliger formula for the smoothing obstruction [Boechat1971].

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

For $k=0,1$ this is covered by Theorem 5.1; for $k\ge2$ it is not.

E.g. by Theorem 5.3 there is a 1-1 correspondence $E^m(S^p\times S^q)\to\pi_p(X)\oplus\pi_q(X)$, $X:=V_{M,M+p+q-m+1}$, for $M>p+q$, $m\ge2q+3$ and $2m\ge3q+3p+4$. For a generalization see Knotted tori [Skopenkov2002].

Some estimations of $E^{2n-k-1}(N)$ for a closed $k$-connected $n$-manifold $N$ are presented in [Skopenkov2010].

5.2 The Whitney invariant

Let $N$ be an $n$-manifold and $f,f_0:N\to\Rr^m$ embeddings. Roughly speaking, $W(f,f_0)=W_{f_0}(f)$ is defined as the homology class of the self-intersection set $\Sigma(H)$ of a general position homotopy $H$ between $f$ and $f_0$. We present an accurate definition in the smooth category for $m\ge n+2$ when either $m-n$ is even or $N$ is orientable [Skopenkov2010]. The definition in the PL category is analogous [Hudson1969], \S12, [Vrabec1977], p. 145, [Skopenkov2006], \S2.4.

Fix orientations on $N$ and on $\Rr^m$. Take embeddings $f,f_0:N\to\Rr^m$. Take a general position homotopy $H:N\times I\to\Rr^m\times I$ between $f_0$ and $f$. By general position the closure $Cl\Sigma(H)$ of the self-intersection set has codimension 2 singularities and so carries a homology class with $\Zz_2$ coefficients. (Note that $Cl\Sigma(H)$ can be assumed to be a submanifold for $2m\ge3n+2$.) For $m-n$ odd it has a natural orientation and so carries a homology class with $\Zz$ coefficients. Define the Whitney invariant

by $W(f)=W_{f_0}(f):=[Cl\Sigma(H)].$ Analogously to [Skopenkov2006], \S2.4, this is well-defined.

6 An orientation on the self-intersection set

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

• (1) $\Sigma(f)$ has a natural orientation.
• (2) the natural orientation on $\Sigma(f)$ need not extend to $Cl\Sigma(f)$.
• (3) the natural orientation on $\Sigma(f)$ extend to $Cl\Sigma(f)$ if $m-n$ is odd [Hudson1969], Lemma 11.4.
• (4) $f\Sigma(f)$ has a natural orientation if $m-n$ is even.

Fix an orientation on $N$ and on $\Rr^m$.

Let us prove (1). Take points $x,y\in N$ outside singularities of $\Sigma(f)$ and such that $fx=fy$. Then a $(2n-m)$-base $\xi_x$ tangent to $\Sigma(f)$ at $x$ gives a $(2n-m)$-base $\xi_y:=df_y^{-1}df_x(\xi_x)$ tangent to $\Sigma(f)$ at $y$. Since $N$ is orientable, we can take positive $(m-n)$-bases $\eta_x$ and $\eta_y$ at $x$ and $y$ normal to $\xi_x$ and to $\xi_y$. If the base $(df_x(\xi_x),df_x(\eta_x),df_y(\eta_y))$ of $\Rr^m$ is positive, then call the base $\xi_x$ positive. This is well-defined because a change of the sign of $\xi_x$ forces changes of the signs of $\xi_y,\eta_x$ and $\eta_y$. (Note that a change of the orientation of $N$ forces changes of the signs of $\eta_x$ and $\eta_y$ and so does not change the orientation of $\Sigma(f)$.)

We can see that (2) holds by considering the cone $D^3\to\Rr^5$ over a general position map $S^2\to\Rr^4$ having only one self-intersection point.

Let us prove (4). Take a $(2n-m)$-base $\xi$ at a point $x\in f\Sigma(f)$ outside singularities of $f\Sigma(f)$. Since $N$ is orientable, we can take a positive $(m-n)$-base $\eta_+$ normal to $f\Sigma(f)$ in one sheet of $f(N)$. Analogously construct an $(m-n)$-base $\eta_-$ for the other sheet of $f(N)$. If $m-n$ is even, then the orientation of the base $(\xi,\eta_+,\eta_-)$ of $\Rr^m$ does not depend on choosing the first and the other sheet of $f(N)$. If the base $(\xi,\eta_+,\eta_-)$ is positive, then call the base $\xi$ positive. This is well-defined because a change of the sign of $\xi$ forces changes of the signs of $\eta_+,\eta_-$ and so of $(\xi,\eta_+,\eta_-)$. (Note that a change of the orientation of $N$ forces changes of the signs of $\eta_+,\eta_-$ and so does not change the orientation of $f\Sigma(f)$.)

7 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
• [Becker&Glover1971] J. Becker and H. Glover, Note on the embedding of manifolds in Euclidean space, Proc. Am. Math. Soc. 27 (1971), 405-410. MR0268903 (42 #3800) Zbl 0207.22402
• [Boechat&Haefliger1970] J. Boéchat and A. Haefliger, Plongements différentiables des variétés orientées de dimension $4$ dans $\Rr^7$, (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 $4k$ dans $\Rr^{6k+1}$, 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
• [Saeki1999] O. Saeki, On punctured 3-manifolds in 5-sphere, Hiroshima Math. J. 29 (1999) 255--272, MR1704247 (2000h:57045)

• [Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
• [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.
• [Skopenkov2008] A. Skopenkov, A classification of smooth embeddings of 3-manifolds in 6-space, Math. Z. 260 (2008), no.3, 647–672. Available at the arXiv:0603429MR2434474 (2010e:57028) Zbl 1167.57013

• [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into $\Rr^{2n-k-1}$, Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
• [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470

• [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

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

\in D^n$ on one component N is a connected manifold of dimension $n>1$, and $m \ge2n+1$, then every two embeddings $N \to\Rr^m$ are isotopic. In this page we summarise the situation for $m=2n\ge6$ and some more general situations.

See general introduction on embeddings, notation and conventions.

2 Classification

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

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

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

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

3 Examples

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

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

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

3.1 Hudson tori

In this subsection we construct, for $a\in\Zz$ and $n\ge2$, an embedding

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

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

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

One guesses that $\Hud_n(a)$ is not isotopic to $\Hud_n(a')$ for $a\ne a'$. And that a $\Zz$-valued invariant exists and is `realized' by the homotopy class of the map

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

Proposition 3.3 follows by calculation of the Whitney invariant (Remark 4.1.e below) and, for $n$ even, by Theorem 2.1. Analogously, $\Hud_2(a)$ is not isotopic to $\Hud_2(a')$ if $a\not\equiv a'\mod2$. It would be interesting to know if the converse holds. E.g. is $\Hud_2(0)$ isotopic to $\Hud_2(2)$? It would also be interesting to find an explicit construction of an isotopy between $\Hud_{2k}(a)$ and $\Hud_{2k}(a+2)$ (cf. [Vrabec1977], \S6).

3.2 Action by linked embedded connected sum

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

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

Take a vector field on $S^1\times*$ normal to $S^1\times S^{n-1}$. Extend $S^1\times*$ along this vector field to a map $\overline a:D^2\to\Rr^{2n}$. Since $2n>4$ 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

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

with natural orientation.

By definition of the Whitney invariant, $W(f_a)$ 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 $H_1(N;\Zz_{(n-1)})\to E^{2n}(N)$.

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

Fix orientations on $\Rr^{2n}$ and, if $n$ is odd, 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 [Hirsch1959]. Since $N_0$ has an $(n-1)$-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 $f$ 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:

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 $f\cap F$.

5 A generalization to highly-connected manifolds

Let $N$ be a closed $k$-connected $n$-manifold. We present description of $E^{2n-k}(N)$ generalizing Theorem 2.1, and its generalization to $E^m(N)$ for $m\ge2n-2k+1$.

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

5.1 Classification

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

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

E.g. by Theorem 5.1 the Whitney invariant $W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{(q)}$ is bijective for $1\le p\le q-2$. It is in fact a group isomorphism; the generator is the Hudson torus.

Because of the existence of knots the analogues of Theorem 5.1 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 Bo\'echat-Haefliger formula for the smoothing obstruction [Boechat1971].

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

For $k=0,1$ this is covered by Theorem 5.1; for $k\ge2$ it is not.

E.g. by Theorem 5.3 there is a 1-1 correspondence $E^m(S^p\times S^q)\to\pi_p(X)\oplus\pi_q(X)$, $X:=V_{M,M+p+q-m+1}$, for $M>p+q$, $m\ge2q+3$ and $2m\ge3q+3p+4$. For a generalization see Knotted tori [Skopenkov2002].

Some estimations of $E^{2n-k-1}(N)$ for a closed $k$-connected $n$-manifold $N$ are presented in [Skopenkov2010].

5.2 The Whitney invariant

Let $N$ be an $n$-manifold and $f,f_0:N\to\Rr^m$ embeddings. Roughly speaking, $W(f,f_0)=W_{f_0}(f)$ is defined as the homology class of the self-intersection set $\Sigma(H)$ of a general position homotopy $H$ between $f$ and $f_0$. We present an accurate definition in the smooth category for $m\ge n+2$ when either $m-n$ is even or $N$ is orientable [Skopenkov2010]. The definition in the PL category is analogous [Hudson1969], \S12, [Vrabec1977], p. 145, [Skopenkov2006], \S2.4.

Fix orientations on $N$ and on $\Rr^m$. Take embeddings $f,f_0:N\to\Rr^m$. Take a general position homotopy $H:N\times I\to\Rr^m\times I$ between $f_0$ and $f$. By general position the closure $Cl\Sigma(H)$ of the self-intersection set has codimension 2 singularities and so carries a homology class with $\Zz_2$ coefficients. (Note that $Cl\Sigma(H)$ can be assumed to be a submanifold for $2m\ge3n+2$.) For $m-n$ odd it has a natural orientation and so carries a homology class with $\Zz$ coefficients. Define the Whitney invariant

by $W(f)=W_{f_0}(f):=[Cl\Sigma(H)].$ Analogously to [Skopenkov2006], \S2.4, this is well-defined.

6 An orientation on the self-intersection set

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

• (1) $\Sigma(f)$ has a natural orientation.
• (2) the natural orientation on $\Sigma(f)$ need not extend to $Cl\Sigma(f)$.
• (3) the natural orientation on $\Sigma(f)$ extend to $Cl\Sigma(f)$ if $m-n$ is odd [Hudson1969], Lemma 11.4.
• (4) $f\Sigma(f)$ has a natural orientation if $m-n$ is even.

Fix an orientation on $N$ and on $\Rr^m$.

Let us prove (1). Take points $x,y\in N$ outside singularities of $\Sigma(f)$ and such that $fx=fy$. Then a $(2n-m)$-base $\xi_x$ tangent to $\Sigma(f)$ at $x$ gives a $(2n-m)$-base $\xi_y:=df_y^{-1}df_x(\xi_x)$ tangent to $\Sigma(f)$ at $y$. Since $N$ is orientable, we can take positive $(m-n)$-bases $\eta_x$ and $\eta_y$ at $x$ and $y$ normal to $\xi_x$ and to $\xi_y$. If the base $(df_x(\xi_x),df_x(\eta_x),df_y(\eta_y))$ of $\Rr^m$ is positive, then call the base $\xi_x$ positive. This is well-defined because a change of the sign of $\xi_x$ forces changes of the signs of $\xi_y,\eta_x$ and $\eta_y$. (Note that a change of the orientation of $N$ forces changes of the signs of $\eta_x$ and $\eta_y$ and so does not change the orientation of $\Sigma(f)$.)

We can see that (2) holds by considering the cone $D^3\to\Rr^5$ over a general position map $S^2\to\Rr^4$ having only one self-intersection point.

Let us prove (4). Take a $(2n-m)$-base $\xi$ at a point $x\in f\Sigma(f)$ outside singularities of $f\Sigma(f)$. Since $N$ is orientable, we can take a positive $(m-n)$-base $\eta_+$ normal to $f\Sigma(f)$ in one sheet of $f(N)$. Analogously construct an $(m-n)$-base $\eta_-$ for the other sheet of $f(N)$. If $m-n$ is even, then the orientation of the base $(\xi,\eta_+,\eta_-)$ of $\Rr^m$ does not depend on choosing the first and the other sheet of $f(N)$. If the base $(\xi,\eta_+,\eta_-)$ is positive, then call the base $\xi$ positive. This is well-defined because a change of the sign of $\xi$ forces changes of the signs of $\eta_+,\eta_-$ and so of $(\xi,\eta_+,\eta_-)$. (Note that a change of the orientation of $N$ forces changes of the signs of $\eta_+,\eta_-$ and so does not change the orientation of $f\Sigma(f)$.)

7 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
• [Becker&Glover1971] J. Becker and H. Glover, Note on the embedding of manifolds in Euclidean space, Proc. Am. Math. Soc. 27 (1971), 405-410. MR0268903 (42 #3800) Zbl 0207.22402
• [Boechat&Haefliger1970] J. Boéchat and A. Haefliger, Plongements différentiables des variétés orientées de dimension $4$ dans $\Rr^7$, (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 $4k$ dans $\Rr^{6k+1}$, 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
• [Saeki1999] O. Saeki, On punctured 3-manifolds in 5-sphere, Hiroshima Math. J. 29 (1999) 255--272, MR1704247 (2000h:57045)

• [Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
• [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.
• [Skopenkov2008] A. Skopenkov, A classification of smooth embeddings of 3-manifolds in 6-space, Math. Z. 260 (2008), no.3, 647–672. Available at the arXiv:0603429MR2434474 (2010e:57028) Zbl 1167.57013

• [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into $\Rr^{2n-k-1}$, Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
• [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470

• [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

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

\times S^{n-1}$ and the `standard inclusion' $\{-1\}\times S^{n-1}\to\partial D^n\subset D^n$ on the other component. This map gives an ''embedding'' $$S^0\times S^{n-1}\to D^n\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$ (See Figure 2.2 of \cite{Skopenkov2006}. The image of this embedding is the union of the standard $S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$ and the graph of the identity map in $S^{n-1}\times S^{n-1}\subset D^n\times S^{n-1}\subset\Rr^{2n-1}\subset\Rr^{2n}$.) Take any $x\in S^{n-1}$. The disk $D^{n+1}\times x$ intersects the image of this embedding by two points lying in $D^n\times x$, i.e., by the image of an embedding $S^0\times x\to D^n\times x$. Extend the latter embedding to an embedding $S^1\times x\to D^{n+1}\times x$. (See Figure 2.3 of \cite{Skopenkov2006}.) Thus we obtain ''the Hudson torus'' $$\Hud_n'(1):S^1\times S^{n-1}\overset h\to D^{n+1}\times S^{n-1}\subset\Rr^{2n}.$$ Here $h(e^{i\theta},y):=(y\cos\theta,\sin\theta,y)$, where $D^{n+1}$ is identified with $D^n\times D^1$. The embedding $\Hud_n'(a)$ is obtained in the same way starting from a map $\varphi:\{-1\}\times S^{n-1}\to\partial D^n$ of degree $a$ instead of the `standard inclusion'. {{endthm}} {{beginthm|Remark}}\label{rhe} (a) The analogue of Proposition \ref{pr3} for $\Hud_n$ replaced to $\Hud_n'$ holds, with analogous proof. (b) Embeddings $\Hud_n(a)$ and $\Hud_n'(a)$ are smoothly isotopic for $n\ge4$ and are PL isotopic for $n\ge3$ \cite{Skopenkov2006a}. It would be interesting to know if they are isotopic for $n=2$, or are smoothly isotopic for $n=3$. (c) For $n=2$ these construction give what we call the ''left'' Hudson torus. The ''right'' Hudson torus is constructed analogously. It is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of $S^1\times S^1$. The right and the left Hudson tori are not isotopic by Remark \ref{re5}.e below. (d) Analogously one constructs the Hudson torus $\Hud_{n,p}(a):S^p\times S^{n-p}\to\Rr^{2n-p+1}$ for $a\in\Zz$ or, more generally, $\Hud_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$ for $a\in\pi_n(S^{m-n+p-1})$. There are versions of these constructions corresponding to Definition \ref{dh2n}. For $p=0$ this corresponds to [[Links,_i.e._embeddings_of_non-connected_manifolds#The_Zeeman_construction_and_linking_coefficient|the Zeeman construction]] and its composition with ''the second unframed Kirby move''. It would be interesting to know if links $\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$ are isotopic, cf. \cite{Skopenkov2015a}, Remark 2.9.b. These constructions could be [[Knotted_tori#Examples|further generalized]]. {{endthm}} ===Action by linked embedded connected sum=== ; In this subsection we generalize the construction of Hudson torus $\Hud(a)$. For $n\ge3$, a closed connected orientable $n$-manifold $N$, an embedding $f_0:N\to\Rr^{2n}$ and $a\in H_1(N)$, we construct an embedding $f_a:N\to\Rr^{2n}$. This embedding is obtained by linked embedded connected sum of $f_0$ with an $n$-sphere representing homology Alexander dual of $a$. More precisely, represent $a$ by an embedding $a:S^1\to N$. Since any orientable bundle over $S^1$ is trivial, $\nu_{f_0}^{-1}a(S^1)\cong S^1\times S^{n-1}$. Identify $\nu_{f_0}^{-1}a(S^1)$ with $S^1\times S^{n-1}$. In the next paragraph we recall definition of embedded surgery of $S^1\times*\subset S^1\times S^{n-1}$ which yields an embedding $g:S^n\to C_{f_0}$. Then we define $f_a$ to be the (linked) embedded connected sum of $f_0$ and $g$ (along certain arc joining their images). Take a vector field on $S^1\times*$ normal to $S^1\times S^{n-1}$. Extend $S^1\times*$ along this vector field to a 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 $$\widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}.$$ Define embedding $g:S^n\to C_{f_0}$ by setting $$g(S^n):\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n$$ with natural orientation. By [[Embeddings_just_below_the_stable_range:_classification#The_Whitney_invariant_.28for_either_n_even_or_N_orientable.29| definition of the Whitney invariant]], $W(f_a)$ 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 even or N orientable) == ; Fix orientations on $\Rr^{2n}$ and, if $n$ is odd, 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: $$W(f):=[Cl(f\cap F)]\in H_1(N_0,\partial N_0;\Zz_{(n-1)})\cong H_1(N;\Zz_{(n-1)}).$$ 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} (a) 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 [[Embeddings just below the stable range: classification#The Whitney invariant|an alternative one]]. It is for being well-defined that we need $\Zz_2$-coefficients when $n$ is even. (b) 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. (c) Since a change of the orientation on $N$ forces a change of the orientation on $B^n$, the class $W(f)$ is independent of the choice of the orientation on $N$. For the reflection $\sigma:\Rr^{2n}\to\Rr^{2n}$ 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 $f\cap F$). (d) The above definition makes sense for each $n$, not only for $n\ge3$. (e) For the [[Embeddings just below the stable range: classification#Hudson tori|Hudson tori]] $W(\Hud_n(a))=W(\Hud'_n(a))$ is $a$ or $a\mod2$ for $n\ge3$, is $(a\mod2,0)$ for $n=2$. (f) $W(f\#g)=W(f)$ for each embeddings $f:N\to\Rr^{2n}$ and $g:S^n\to\Rr^{2n}$. {{endthm}} == A generalization to highly-connected manifolds == ; Let $N$ be a closed $k$-connected $n$-manifold. We present description of $E^{2n-k}(N)$ generalizing Theorem \ref{th4}, and its generalization to $E^m(N)$ for $m\ge2n-2k+1$. Examples are [[Embeddings just below the stable range: classification#Remarks|Hudson tori]] $\Hud_{m,n,p}(a), \Hud'_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$. === Classification === ; {{beginthm|Theorem}}\label{hico} Let $N$ be a closed orientable homologically $k$-connected $n$-manifold, $k\ge0$. Then [[Embeddings just below the stable range: classification#The Whitney invariant|the Whitney invariant]] $$W:E^{2n-k}(N)\to H_{k+1}(N,\Zz_{(n-k-1)})$$ is a bijection, provided $n\ge k+3$ or $n\ge2k+4$ in the PL or DIFF categories, respectively. {{endthm}} Theorem \ref{hico} was proved in \cite{Haefliger&Hirsch1963}, \cite{Hudson1969}, \S11, \cite{Boechat&Haefliger1970}, \cite{Boechat1971}, \cite{Vrabec1977} ''homotopically'' $k$-connected manifolds. The proof works for ''homologically'' $k$-connected manifolds. For $k=0$ this is covered by Theorem \ref{th4}; for $k\ge1$ it is not. The PL case of Theorem \ref{hico} gives nothing but [[High codimension embeddings: classification#Unknotting theorems|the Unknotting Spheres Theorem]] for $k+3\le n\le2k+1$. E.g. by Theorem \ref{hico} the Whitney invariant $W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{(q)}$ is bijective for N is a connected manifold of dimension $n>1$, and $m \ge2n+1$, then every two embeddings $N \to\Rr^m$ are isotopic. In this page we summarise the situation for $m=2n\ge6$ and some more general situations.

See general introduction on embeddings, notation and conventions.

2 Classification

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

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

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

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

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

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

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

3 Examples

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

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

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

3.1 Hudson tori

In this subsection we construct, for $a\in\Zz$ and $n\ge2$, an embedding

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

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

Definition 3.1. (This construction, as opposed to Definition 3.2, works for $n=1$.) Take the standard embeddings $2D^{n+1}\times S^{n-1}\subset\Rr^{2n}$ (where $2$ means homothety with coefficient 2) and $\partial D^2\subset\partial D^{n+1}$. Take embedded sphere and embedded torus

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

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

Definition 3.2. For $a\in\Zz$ instead of $2\partial D^{n+1}\times*$ we take $|a|$ copies $(1+\frac1k)\partial D^{n+1}\times*$ ($k=1,\dots,|a|$) of $n$-sphere outside $D^{n+1}\times S^{n-1}$ `parallel' to $\partial D^{n+1}\times *$, with standard orientation for $a>0$ or the opposite orientation for $a<0$. Then we make embedded connected sum by tubes joining each $k$-th copy to $(k+1)$-th copy. We obtain an embedding $g:S^n\to\Rr^{2n}$. Let $\Hud_n(a)$ be the linked embedded connected sum of $g$ with the embedding $\partial D^2\times S^{n-1}\subset\Rr^{2n}$ from Definition 3.1.

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

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

One guesses that $\Hud_n(a)$ is not isotopic to $\Hud_n(a')$ for $a\ne a'$. And that a $\Zz$-valued invariant exists and is `realized' by the homotopy class of the map

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

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

Proposition 3.3. 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 $a\equiv a'\mod2$.

Proposition 3.3 follows by calculation of the Whitney invariant (Remark 4.1.e below) and, for $n$ even, by Theorem 2.1. Analogously, $\Hud_2(a)$ is not isotopic to $\Hud_2(a')$ if $a\not\equiv a'\mod2$. It would be interesting to know if the converse holds. E.g. is $\Hud_2(0)$ isotopic to $\Hud_2(2)$? It would also be interesting to find an explicit construction of an isotopy between $\Hud_{2k}(a)$ and $\Hud_{2k}(a+2)$ (cf. [Vrabec1977], \S6).

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

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

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

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

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

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

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

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

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

Remark 3.5. (a) The analogue of Proposition 3.3 for $\Hud_n$ replaced to $\Hud_n'$ holds, with analogous proof.

(b) Embeddings $\Hud_n(a)$ and $\Hud_n'(a)$ are smoothly isotopic for $n\ge4$ and are PL isotopic for $n\ge3$ [Skopenkov2006a]. It would be interesting to know if they are isotopic for $n=2$, or are smoothly isotopic for $n=3$.

(c) For $n=2$ these construction give what we call the left Hudson torus. The right Hudson torus is constructed analogously. It is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of $S^1\times S^1$. The right and the left Hudson tori are not isotopic by Remark 4.1.e below.

(d) Analogously one constructs the Hudson torus $\Hud_{n,p}(a):S^p\times S^{n-p}\to\Rr^{2n-p+1}$ for $a\in\Zz$ or, more generally, $\Hud_{m,n,p}(a):S^p\times S^{n-p}\to\Rr^m$ for $a\in\pi_n(S^{m-n+p-1})$. There are versions of these constructions corresponding to Definition 3.4. For $p=0$ this corresponds to the Zeeman construction and its composition with the second unframed Kirby move. It would be interesting to know if links $\Hud_{1,0}(a),\Hud'_{1,0}(a):S^0\times S^1\to\Rr^3$ are isotopic, cf. [Skopenkov2015a], Remark 2.9.b. These constructions could be further generalized.

3.2 Action by linked embedded connected sum

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

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

Take a vector field on $S^1\times*$ normal to $S^1\times S^{n-1}$. Extend $S^1\times*$ along this vector field to a map $\overline a:D^2\to\Rr^{2n}$. Since $2n>4$ 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

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

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

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

with natural orientation.

By definition of the Whitney invariant, $W(f_a)$ 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 $H_1(N;\Zz_{(n-1)})\to E^{2n}(N)$.

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

Fix orientations on $\Rr^{2n}$ and, if $n$ is odd, 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 [Hirsch1959]. Since $N_0$ has an $(n-1)$-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 $f$ 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:

$$\displaystyle W(f):=[Cl(f\cap F)]\in H_1(N_0,\partial N_0;\Zz_{(n-1)})\cong H_1(N;\Zz_{(n-1)}).$$

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 $f\cap F$.

Remark 4.1. (a) 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 an alternative one. It is for being well-defined that we need $\Zz_2$-coefficients when $n$ is even.

(b) 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.

(c) Since a change of the orientation on $N$ forces a change of the orientation on $B^n$, the class $W(f)$ is independent of the choice of the orientation on $N$. For the reflection $\sigma:\Rr^{2n}\to\Rr^{2n}$ 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 $f\cap F$).

(d) The above definition makes sense for each $n$, not only for $n\ge3$.

(e) For the Hudson tori $W(\Hud_n(a))=W(\Hud'_n(a))$ is $a$ or $a\mod2$ for $n\ge3$, is $(a\mod2,0)$ for $n=2$.

(f) $W(f\#g)=W(f)$ for each embeddings $f:N\to\Rr^{2n}$ and $g:S^n\to\Rr^{2n}$.

5 A generalization to highly-connected manifolds

Let $N$ be a closed $k$-connected $n$-manifold. We present description of $E^{2n-k}(N)$ generalizing Theorem 2.1, and its generalization to $E^m(N)$ for $m\ge2n-2k+1$.

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

5.1 Classification

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

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

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

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

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

E.g. by Theorem 5.1 the Whitney invariant $W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{(q)}$ is bijective for $1\le p\le q-2$. It is in fact a group isomorphism; the generator is the Hudson torus.

Because of the existence of knots the analogues of Theorem 5.1 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 Bo\'echat-Haefliger formula for the smoothing obstruction [Boechat1971].

Theorem 5.2. [Skopenkov2008] Let $N$ be a closed homologically $(2k-2)$-connected $(4k-1)$-manifold. Then the Whitney invariant

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

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

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

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

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

Theorem 5.3. [Becker&Glover1971] Let $N$ be a closed $k$-connected $n$-manifold embeddable into $\Rr^m$, $m\ge2n-2k+1$ and $2m\ge 3n+4$. Then there is a 1-1 correspondence

$$\displaystyle E^m(N)\to [N_0;V_{M,M+n-m+1}],\quad\mbox{where}\quad M>n.$$

For $k=0,1$ this is covered by Theorem 5.1; for $k\ge2$ it is not.

E.g. by Theorem 5.3 there is a 1-1 correspondence $E^m(S^p\times S^q)\to\pi_p(X)\oplus\pi_q(X)$, $X:=V_{M,M+p+q-m+1}$, for $M>p+q$, $m\ge2q+3$ and $2m\ge3q+3p+4$. For a generalization see Knotted tori [Skopenkov2002].

Some estimations of $E^{2n-k-1}(N)$ for a closed $k$-connected $n$-manifold $N$ are presented in [Skopenkov2010].

5.2 The Whitney invariant

Let $N$ be an $n$-manifold and $f,f_0:N\to\Rr^m$ embeddings. Roughly speaking, $W(f,f_0)=W_{f_0}(f)$ is defined as the homology class of the self-intersection set $\Sigma(H)$ of a general position homotopy $H$ between $f$ and $f_0$. We present an accurate definition in the smooth category for $m\ge n+2$ when either $m-n$ is even or $N$ is orientable [Skopenkov2010]. The definition in the PL category is analogous [Hudson1969], \S12, [Vrabec1977], p. 145, [Skopenkov2006], \S2.4.

Fix orientations on $N$ and on $\Rr^m$. Take embeddings $f,f_0:N\to\Rr^m$. Take a general position homotopy $H:N\times I\to\Rr^m\times I$ between $f_0$ and $f$. By general position the closure $Cl\Sigma(H)$ of the self-intersection set has codimension 2 singularities and so carries a homology class with $\Zz_2$ coefficients. (Note that $Cl\Sigma(H)$ can be assumed to be a submanifold for $2m\ge3n+2$.) For $m-n$ odd it has a natural orientation and so carries a homology class with $\Zz$ coefficients. Define the Whitney invariant

$$\displaystyle W:E^m(N)\to H_{2n-m+1}(N\times I;\Zz_{(m-n-1)})\cong H_{2n-m+1}(N;\Zz_{(m-n-1)})$$

by $W(f)=W_{f_0}(f):=[Cl\Sigma(H)].$ Analogously to [Skopenkov2006], \S2.4, this is well-defined.

6 An orientation on the self-intersection set

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

• (1) $\Sigma(f)$ has a natural orientation.
• (2) the natural orientation on $\Sigma(f)$ need not extend to $Cl\Sigma(f)$.
• (3) the natural orientation on $\Sigma(f)$ extend to $Cl\Sigma(f)$ if $m-n$ is odd [Hudson1969], Lemma 11.4.
• (4) $f\Sigma(f)$ has a natural orientation if $m-n$ is even.

Fix an orientation on $N$ and on $\Rr^m$.

Let us prove (1). Take points $x,y\in N$ outside singularities of $\Sigma(f)$ and such that $fx=fy$. Then a $(2n-m)$-base $\xi_x$ tangent to $\Sigma(f)$ at $x$ gives a $(2n-m)$-base $\xi_y:=df_y^{-1}df_x(\xi_x)$ tangent to $\Sigma(f)$ at $y$. Since $N$ is orientable, we can take positive $(m-n)$-bases $\eta_x$ and $\eta_y$ at $x$ and $y$ normal to $\xi_x$ and to $\xi_y$. If the base $(df_x(\xi_x),df_x(\eta_x),df_y(\eta_y))$ of $\Rr^m$ is positive, then call the base $\xi_x$ positive. This is well-defined because a change of the sign of $\xi_x$ forces changes of the signs of $\xi_y,\eta_x$ and $\eta_y$. (Note that a change of the orientation of $N$ forces changes of the signs of $\eta_x$ and $\eta_y$ and so does not change the orientation of $\Sigma(f)$.)

We can see that (2) holds by considering the cone $D^3\to\Rr^5$ over a general position map $S^2\to\Rr^4$ having only one self-intersection point.

Let us prove (4). Take a $(2n-m)$-base $\xi$ at a point $x\in f\Sigma(f)$ outside singularities of $f\Sigma(f)$. Since $N$ is orientable, we can take a positive $(m-n)$-base $\eta_+$ normal to $f\Sigma(f)$ in one sheet of $f(N)$. Analogously construct an $(m-n)$-base $\eta_-$ for the other sheet of $f(N)$. If $m-n$ is even, then the orientation of the base $(\xi,\eta_+,\eta_-)$ of $\Rr^m$ does not depend on choosing the first and the other sheet of $f(N)$. If the base $(\xi,\eta_+,\eta_-)$ is positive, then call the base $\xi$ positive. This is well-defined because a change of the sign of $\xi$ forces changes of the signs of $\eta_+,\eta_-$ and so of $(\xi,\eta_+,\eta_-)$. (Note that a change of the orientation of $N$ forces changes of the signs of $\eta_+,\eta_-$ and so does not change the orientation of $f\Sigma(f)$.)

7 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
• [Becker&Glover1971] J. Becker and H. Glover, Note on the embedding of manifolds in Euclidean space, Proc. Am. Math. Soc. 27 (1971), 405-410. MR0268903 (42 #3800) Zbl 0207.22402
• [Boechat&Haefliger1970] J. Boéchat and A. Haefliger, Plongements différentiables des variétés orientées de dimension $4$ dans $\Rr^7$, (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 $4k$ dans $\Rr^{6k+1}$, 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
• [Saeki1999] O. Saeki, On punctured 3-manifolds in 5-sphere, Hiroshima Math. J. 29 (1999) 255--272, MR1704247 (2000h:57045)

• [Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
• [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.
• [Skopenkov2008] A. Skopenkov, A classification of smooth embeddings of 3-manifolds in 6-space, Math. Z. 260 (2008), no.3, 647–672. Available at the arXiv:0603429MR2434474 (2010e:57028) Zbl 1167.57013

• [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into $\Rr^{2n-k-1}$, Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
• [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470

• [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

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

\le p\le q-2$. It is in fact a group isomorphism; the generator is [[Embeddings just below the stable range: classification#Hudson tori|the Hudson torus]]. 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 Bo\'echat-Haefliger formula for the smoothing obstruction \cite{Boechat1971}. {{beginthm|Theorem}}\label{hicod} \cite{Skopenkov2008} Let $N$ be a closed homologically $(2k-2)$-connected $(4k-1)$-manifold. Then [[Embeddings just below the stable range: classification#The Whitney invariant|the Whitney invariant]] $$W:E^{6k}_D(N)\to H_{2k-1}(N)$$ is surjective and for each $u\in H_{2k-1}(N)$ [[3-manifolds in 6-space#The Kreck invariant|the Kreck invariant]] $$\eta_u:W^{-1}u\to\Zz_{d(u)}$$ is a 1-1 correspondence, where $d(u)$ is the divisibility of the projection of $u$ to the free part of $H_1(N)$. {{endthm}} Recall that the divisibility of zero is zero and the divisibility of $x\in G-\{0\}$ is $\max\{d\in\Zz\ | \ \text{there is }x_1\in G: \ x=dx_1\}$. E.g. by Theorem \ref{hicod} the Whitney invariant $W:E^{6k}_D(S^{2k-1}\times S^{2k})\to\Zz$ is surjective and for each $u\in\Zz$ there is a 1-1 correspondence $W^{-1}u\to\Zz_u$. {{beginthm|Theorem}}\label{begl} \cite{Becker&Glover1971} Let $N$ be a closed $k$-connected $n$-manifold embeddable into $\Rr^m$, $m\ge2n-2k+1$ and m\ge 3n+4$. Then there is a 1-1 correspondence $$E^m(N)\to [N_0;V_{M,M+n-m+1}],\quad\mbox{where}\quad M>n.$$ {{endthm}} For $k=0,1$ this is covered by Theorem \ref{hico}; for $k\ge2$ it is not. E.g. by Theorem \ref{begl} there is a 1-1 correspondence $E^m(S^p\times S^q)\to\pi_p(X)\oplus\pi_q(X)$, $X:=V_{M,M+p+q-m+1}$, for $M>p+q$, $m\ge2q+3$ and m\ge3q+3p+4$. For a generalization see [[Knotted tori|Knotted tori]] \cite{Skopenkov2002}. Some estimations of $E^{2n-k-1}(N)$ for a closed $k$-connected $n$-manifold $N$ are presented in \cite{Skopenkov2010}. === The Whitney invariant === ; Let $N$ be an $n$-manifold and $f,f_0:N\to\Rr^m$ embeddings. Roughly speaking, $W(f,f_0)=W_{f_0}(f)$ is defined as the homology class of the self-intersection set $\Sigma(H)$ of a general position homotopy $H$ between $f$ and $f_0$. We present an accurate definition in the smooth category for $m\ge n+2$ when either $m-n$ is even or $N$ is orientable \cite{Skopenkov2010}. The definition in the PL category is analogous \cite{Hudson1969}, \S12, \cite{Vrabec1977}, p. 145, \cite{Skopenkov2006}, \S2.4. Fix orientations on $N$ and on $\Rr^m$. Take embeddings $f,f_0:N\to\Rr^m$. Take a general position homotopy $H:N\times I\to\Rr^m\times I$ between $f_0$ and $f$. By general position the closure $Cl\Sigma(H)$ of the self-intersection set has codimension 2 singularities and so carries a homology class with $\Zz_2$ coefficients. (Note that $Cl\Sigma(H)$ can be assumed to be a submanifold for m\ge3n+2$.) For $m-n$ odd it has a [[Embeddings just below the stable range: classification#An orientation on the self-intersection set|natural orientation]] and so carries a homology class with $\Zz$ coefficients. Define the Whitney invariant $$W:E^m(N)\to H_{2n-m+1}(N\times I;\Zz_{(m-n-1)})\cong H_{2n-m+1}(N;\Zz_{(m-n-1)})$$ by $W(f)=W_{f_0}(f):=[Cl\Sigma(H)].$ Analogously to \cite{Skopenkov2006}, \S2.4, this is well-defined. == An orientation on the self-intersection set == ; Let $f:N\to\Rr^m$ be a general position smooth map of an orientable $n$-manifold $N$. Assume that $m\ge n+2$ so that the closure $Cl\Sigma(f)$ of the self-intersection set of $f$ has codimension 2 singularities. Then * (1) $\Sigma(f)$ has a natural orientation. * (2) the natural orientation on $\Sigma(f)$ need not extend to $Cl\Sigma(f)$. * (3) the natural orientation on $\Sigma(f)$ extend to $Cl\Sigma(f)$ if $m-n$ is odd \cite{Hudson1969}, Lemma 11.4. * (4) $f\Sigma(f)$ has a natural orientation if $m-n$ is even. Fix an orientation on $N$ and on $\Rr^m$. Let us prove (1). Take points $x,y\in N$ outside singularities of $\Sigma(f)$ and such that $fx=fy$. Then a $(2n-m)$-base $\xi_x$ tangent to $\Sigma(f)$ at $x$ gives a $(2n-m)$-base $\xi_y:=df_y^{-1}df_x(\xi_x)$ tangent to $\Sigma(f)$ at $y$. Since $N$ is orientable, we can take positive $(m-n)$-bases $\eta_x$ and $\eta_y$ at $x$ and $y$ normal to $\xi_x$ and to $\xi_y$. If the base $(df_x(\xi_x),df_x(\eta_x),df_y(\eta_y))$ of $\Rr^m$ is positive, then call the base $\xi_x$ positive. This is well-defined because a change of the sign of $\xi_x$ forces changes of the signs of $\xi_y,\eta_x$ and $\eta_y$. (Note that a change of the orientation of $N$ forces changes of the signs of $\eta_x$ and $\eta_y$ and so does not change the orientation of $\Sigma(f)$.) We can see that (2) holds by considering the cone $D^3\to\Rr^5$ over a general position map $S^2\to\Rr^4$ having only one self-intersection point. Let us prove (4). Take a $(2n-m)$-base $\xi$ at a point $x\in f\Sigma(f)$ outside singularities of $f\Sigma(f)$. Since $N$ is orientable, we can take a positive $(m-n)$-base $\eta_+$ normal to $f\Sigma(f)$ in one sheet of $f(N)$. Analogously construct an $(m-n)$-base $\eta_-$ for the other sheet of $f(N)$. If $m-n$ is even, then the orientation of the base $(\xi,\eta_+,\eta_-)$ of $\Rr^m$ does not depend on choosing the first and the other sheet of $f(N)$. If the base $(\xi,\eta_+,\eta_-)$ is positive, then call the base $\xi$ positive. This is well-defined because a change of the sign of $\xi$ forces changes of the signs of $\eta_+,\eta_-$ and so of $(\xi,\eta_+,\eta_-)$. (Note that a change of the orientation of $N$ forces changes of the signs of $\eta_+,\eta_-$ and so does not change the orientation of $f\Sigma(f)$.) == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]]N is a connected manifold of dimension $n>1$, and $m \ge2n+1$, then every two embeddings $N \to\Rr^m$ are isotopic. In this page we summarise the situation for $m=2n\ge6$ and some more general situations.

See general introduction on embeddings, notation and conventions.

2 Classification

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

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

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

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

3 Examples

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

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

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

3.1 Hudson tori

In this subsection we construct, for $a\in\Zz$ and $n\ge2$, an embedding

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

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

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

One guesses that $\Hud_n(a)$ is not isotopic to $\Hud_n(a')$ for $a\ne a'$. And that a $\Zz$-valued invariant exists and is `realized' by the homotopy class of the map

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

Proposition 3.3 follows by calculation of the Whitney invariant (Remark 4.1.e below) and, for $n$ even, by Theorem 2.1. Analogously, $\Hud_2(a)$ is not isotopic to $\Hud_2(a')$ if $a\not\equiv a'\mod2$. It would be interesting to know if the converse holds. E.g. is $\Hud_2(0)$ isotopic to $\Hud_2(2)$? It would also be interesting to find an explicit construction of an isotopy between $\Hud_{2k}(a)$ and $\Hud_{2k}(a+2)$ (cf. [Vrabec1977], \S6).

3.2 Action by linked embedded connected sum

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

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

Take a vector field on $S^1\times*$ normal to $S^1\times S^{n-1}$. Extend $S^1\times*$ along this vector field to a map $\overline a:D^2\to\Rr^{2n}$. Since $2n>4$ 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

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

with natural orientation.

By definition of the Whitney invariant, $W(f_a)$ 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 $H_1(N;\Zz_{(n-1)})\to E^{2n}(N)$.

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

Fix orientations on $\Rr^{2n}$ and, if $n$ is odd, 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 [Hirsch1959]. Since $N_0$ has an $(n-1)$-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 $f$ 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:

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 $f\cap F$.

5 A generalization to highly-connected manifolds

Let $N$ be a closed $k$-connected $n$-manifold. We present description of $E^{2n-k}(N)$ generalizing Theorem 2.1, and its generalization to $E^m(N)$ for $m\ge2n-2k+1$.

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

5.1 Classification

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

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

E.g. by Theorem 5.1 the Whitney invariant $W:E^{p+2q+1}(S^p\times S^q)\to\Zz_{(q)}$ is bijective for $1\le p\le q-2$. It is in fact a group isomorphism; the generator is the Hudson torus.

Because of the existence of knots the analogues of Theorem 5.1 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 Bo\'echat-Haefliger formula for the smoothing obstruction [Boechat1971].

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

For $k=0,1$ this is covered by Theorem 5.1; for $k\ge2$ it is not.

E.g. by Theorem 5.3 there is a 1-1 correspondence $E^m(S^p\times S^q)\to\pi_p(X)\oplus\pi_q(X)$, $X:=V_{M,M+p+q-m+1}$, for $M>p+q$, $m\ge2q+3$ and $2m\ge3q+3p+4$. For a generalization see Knotted tori [Skopenkov2002].

Some estimations of $E^{2n-k-1}(N)$ for a closed $k$-connected $n$-manifold $N$ are presented in [Skopenkov2010].

5.2 The Whitney invariant

Let $N$ be an $n$-manifold and $f,f_0:N\to\Rr^m$ embeddings. Roughly speaking, $W(f,f_0)=W_{f_0}(f)$ is defined as the homology class of the self-intersection set $\Sigma(H)$ of a general position homotopy $H$ between $f$ and $f_0$. We present an accurate definition in the smooth category for $m\ge n+2$ when either $m-n$ is even or $N$ is orientable [Skopenkov2010]. The definition in the PL category is analogous [Hudson1969], \S12, [Vrabec1977], p. 145, [Skopenkov2006], \S2.4.

Fix orientations on $N$ and on $\Rr^m$. Take embeddings $f,f_0:N\to\Rr^m$. Take a general position homotopy $H:N\times I\to\Rr^m\times I$ between $f_0$ and $f$. By general position the closure $Cl\Sigma(H)$ of the self-intersection set has codimension 2 singularities and so carries a homology class with $\Zz_2$ coefficients. (Note that $Cl\Sigma(H)$ can be assumed to be a submanifold for $2m\ge3n+2$.) For $m-n$ odd it has a natural orientation and so carries a homology class with $\Zz$ coefficients. Define the Whitney invariant

by $W(f)=W_{f_0}(f):=[Cl\Sigma(H)].$ Analogously to [Skopenkov2006], \S2.4, this is well-defined.

6 An orientation on the self-intersection set

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

• (1) $\Sigma(f)$ has a natural orientation.
• (2) the natural orientation on $\Sigma(f)$ need not extend to $Cl\Sigma(f)$.
• (3) the natural orientation on $\Sigma(f)$ extend to $Cl\Sigma(f)$ if $m-n$ is odd [Hudson1969], Lemma 11.4.
• (4) $f\Sigma(f)$ has a natural orientation if $m-n$ is even.

Fix an orientation on $N$ and on $\Rr^m$.

Let us prove (1). Take points $x,y\in N$ outside singularities of $\Sigma(f)$ and such that $fx=fy$. Then a $(2n-m)$-base $\xi_x$ tangent to $\Sigma(f)$ at $x$ gives a $(2n-m)$-base $\xi_y:=df_y^{-1}df_x(\xi_x)$ tangent to $\Sigma(f)$ at $y$. Since $N$ is orientable, we can take positive $(m-n)$-bases $\eta_x$ and $\eta_y$ at $x$ and $y$ normal to $\xi_x$ and to $\xi_y$. If the base $(df_x(\xi_x),df_x(\eta_x),df_y(\eta_y))$ of $\Rr^m$ is positive, then call the base $\xi_x$ positive. This is well-defined because a change of the sign of $\xi_x$ forces changes of the signs of $\xi_y,\eta_x$ and $\eta_y$. (Note that a change of the orientation of $N$ forces changes of the signs of $\eta_x$ and $\eta_y$ and so does not change the orientation of $\Sigma(f)$.)

We can see that (2) holds by considering the cone $D^3\to\Rr^5$ over a general position map $S^2\to\Rr^4$ having only one self-intersection point.

Let us prove (4). Take a $(2n-m)$-base $\xi$ at a point $x\in f\Sigma(f)$ outside singularities of $f\Sigma(f)$. Since $N$ is orientable, we can take a positive $(m-n)$-base $\eta_+$ normal to $f\Sigma(f)$ in one sheet of $f(N)$. Analogously construct an $(m-n)$-base $\eta_-$ for the other sheet of $f(N)$. If $m-n$ is even, then the orientation of the base $(\xi,\eta_+,\eta_-)$ of $\Rr^m$ does not depend on choosing the first and the other sheet of $f(N)$. If the base $(\xi,\eta_+,\eta_-)$ is positive, then call the base $\xi$ positive. This is well-defined because a change of the sign of $\xi$ forces changes of the signs of $\eta_+,\eta_-$ and so of $(\xi,\eta_+,\eta_-)$. (Note that a change of the orientation of $N$ forces changes of the signs of $\eta_+,\eta_-$ and so does not change the orientation of $f\Sigma(f)$.)

7 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
• [Becker&Glover1971] J. Becker and H. Glover, Note on the embedding of manifolds in Euclidean space, Proc. Am. Math. Soc. 27 (1971), 405-410. MR0268903 (42 #3800) Zbl 0207.22402
• [Boechat&Haefliger1970] J. Boéchat and A. Haefliger, Plongements différentiables des variétés orientées de dimension $4$ dans $\Rr^7$, (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 $4k$ dans $\Rr^{6k+1}$, 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
• [Saeki1999] O. Saeki, On punctured 3-manifolds in 5-sphere, Hiroshima Math. J. 29 (1999) 255--272, MR1704247 (2000h:57045)

• [Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
• [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.
• [Skopenkov2008] A. Skopenkov, A classification of smooth embeddings of 3-manifolds in 6-space, Math. Z. 260 (2008), no.3, 647–672. Available at the arXiv:0603429MR2434474 (2010e:57028) Zbl 1167.57013

• [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into $\Rr^{2n-k-1}$, Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
• [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470

• [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

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