Embeddings just below the stable range: classification

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1 Introduction

For notation and conventions see high codimension embeddings.

Classification Theorem 1.1. Let $N$$== Introduction == <wikitex>; Together with the Haefliger knotted sphere $S^{2l-1}\to S^{3l}$, the Hudson example was the first example of an embedding in codimension greater than 2 non-isotopic to the standard embedding. (Hudson's construction \cite{Hudson1963} was not as explicit as the one below). For $m\ge n+2$ define the $standard$ $embedding$ $S^p\times S^{n-p}\to S^m$ as the composition $S^p\times S^{n-p}\to\Rr^{p+1}\times\Rr^{n-p+1}\to\Rr^m\to S^m$ of standard embeddings. For notation and conventions see [[High_codimension_embeddings|high codimension embeddings]]. </wikitex> == Examples == === Hudson tori <wikitex>$\Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}$, $a\in\Zz$, $n\ge2$</wikitex> === <wikitex>; Take the standard embeddings D^{n+1}\times S^{n-1}\subset\Rr^{2n}$ (where $ means homothety with coefficient 2) and $\partial D^2\subset\partial D^{n+1}$. Fix a point $x\in S^{n-1}$. The Hudson torus $\Hud_n(1)$ is the embedded connected sum of $\partial D^{n+1}\times x\quad\text{with}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset 2D^{n+1} \times S^{n-1}\subset\Rr^{2n}.$$ (Unlike the connected sum of the subsection [[some general remarks]] this is a `linked' connected sum, i.e. connected sum of two embeddings whose images are not contained in disjoint cubes.) For $a\in\Zz$ instead of an embedded $n$-sphere \partial D^{n+1}\times x$ we can take $a$ copies $(1+\frac1k)\partial D^{n+1}\times x$ ($k=1,\dots,a$) of $n$-sphere outside $D^{n+1}\times S^{n-1}$ `parallel' to $\partial D^{n+1}\times x$. Then we join these spheres by tubes so that the homotopy class of the resulting embedding $$S^n\to S^{2n}-D^{n+1}\times S^{n-1}\simeq S^{2n}-S^{n-1}\simeq S^n \quad\text{will be}\quad a\in\pi_n(S^n)\cong\Zz.$$ Let $\Hud_n(a)$ be the connected sum of this embedding with the above standard embedding $\partial D^2\times S^{n-1}\subset\Rr^{2n}$. Clearly, $\Hud_n(0)$ is isotopic to the standard embedding. {{beginthm|Proposition|}}\label{pr3} For $n\ge3$ odd $\Hud_n(a)$ is isotopic to $\Hud_n(a')$ if and only if $a=a'$. For $n\ge4$ even $\Hud_n(a)$ is isotopic to $\Hud_n(a')$ if and only if $a\equiv a'\mod2$. {{endthm}}. In particular, $\Hud_n(1)$ is not isotopic to $\Hud_n(0)$ for each $n$ (this was the original motivation for Hudson). Proposition \ref{pr3} follows by Remark 5.e and, for $n$ even, by Theorem 4, both results from [[classification just below the stable range]]. It would be interesting to find an explicit construction of an isotopy between $\Hud_{2k}(a)$ and $\Hud_{2k}(a+2)$ (cf. \cite{Vrabec1977}, \S6) and to prove the analogue of Proposition \ref{pr3} for $n=2$. </wikitex> === A different construction: Hudson tori <wikitex> $\Hud_n'(a):S^1\times S^{n-1}\to\Rr^{2n}$, $a\in\Zz$, $n\ge2$ </wikitex>=== <wikitex>; Define a map $S^0\times S^{n-1}\to D^n$ to be the constant <script type="math/tex">N$ be a closed connected $n$$n$-manifold. The Whitney invariant

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

is bijective if either $n\ge4$$n\ge4$ or $n=3$$n=3$ and CAT=PL [Haefliger&Hirsch1963], [Bausum1975], [Vrabec1977], cf. [Hudson1969]

Classification of smooth embeddings of 3-manifolds in the 6-space is more complicated.

2 Definition of the Whitney invariant (for either n odd or N orientable)

Fix orientations on $\Rr^{2n}$$\Rr^{2n}$ and, if $N$$N$ is even, on $N$$N$. Fix an embedding $f_0:N\to\Rr^{2n}$$f_0:N\to\Rr^{2n}$. For an embedding $f:N\to\Rr^{2n}$$f:N\to\Rr^{2n}$ the restrictions of $f$$f$ and $f_0$$f_0$ to $N_0$$N_0$ are regular homotopic [Hirsch1959]. Since $N_0$$N_0$ has an $(n-1)$$(n-1)$-dimensional spine, it follows that these restrictions are isotopic, cf. [Haefliger&Hirsch1963], 3.1.b, [Takase2006], Lemma 2.2. So we can make an isotopy of $f$$f$ and assume that $f=f_0$$f=f_0$ on $N_0$$N_0$. Take a general position homotopy $F:B^n\times I\to\Rr^{2n}$$F:B^n\times I\to\Rr^{2n}$ relative to $\partial B^n$$\partial B^n$ between the restrictions of $f$$f$ and $g$$g$ to $B^n$$B^n$. Then $f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$$f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$ (i.e. `the intersection of this homotopy with $f(N-B^n)$$f(N-B^n)$') is a 1-manifold (possibly non-compact) without boundary. Define $W(f)$$W(f)$ to be the homology class of the closure of this 1-manifold:

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

The orientation on $f\cap F$$f\cap F$ is defined for $N$$N$ orientable as follows. (This orientation is defined for each $n$$n$ but used only for odd $n$$n$.) For each point $x\in f\cap F$$x\in f\cap F$ take a vector at $x$$x$ tangent to $f\cap F$$f\cap F$. Complete this vector to a positive base tangent to $N$$N$. Since $n+2(n+1)>2\cdot2n$$n+2(n+1)>2\cdot2n$, by general position there is a unique point $y\in B^n\times I$$y\in B^n\times I$ such that $Fy=fx$$Fy=fx$. The tangent vector at $x$$x$ thus gives a tangent vector at $y$$y$ to $B^n\times I$$B^n\times I$. Complete this vector to a positive base tangent to $B^n\times I$$B^n\times I$, where the orientation on $B^n$$B^n$ comes from $N$$N$. The union of the images of the constructed two bases is a base at $Fy=fx$$Fy=fx$ of $\Rr^{2n}$$\Rr^{2n}$. If this base is positive, then call the initial vector of $f\cap F$$f\cap F$ positive. Since a change of the orientation on $f\cap F$$f\cap F$ forces a change of the orientation of the latter base of $\Rr^{2n}$$\Rr^{2n}$, it follows that this condition indeed defines an orientation on $f\cap F$$f\cap F$.

Remark 2.1.

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

Construction of the embedding f a:N\to\Rr^{2n} from an embedding f 0:N\to\Rr^{2n} and a\in H 1(N) (for orientable N and n\ge3) 3 Construction of the embedding $f_a:N\to\Rr^{2n}$$f_a:N\to\Rr^{2n}$ from an embedding $f_0:N\to\Rr^{2n}$$f_0:N\to\Rr^{2n}$ and $a\in H_1(N)$$a\in H_1(N)$ (for orientable $N$$N$ and $n\ge3$$n\ge3$)

Represent $a$$a$ by an embedding $a:S^1\to N$$a:S^1\to N$. Since any orientable bundle over $S^1$$S^1$ is trivial, $\nu_{f_0}^{-1}a(S^1)\cong S^1\times S^{n-1}$$\nu_{f_0}^{-1}a(S^1)\cong S^1\times S^{n-1}$. Identify $\nu_{f_0}^{-1}a(S^1)$$\nu_{f_0}^{-1}a(S^1)$ with $S^1\times S^{n-1}$$S^1\times S^{n-1}$. It remains to make an embedded surgery of $S^1\times*\subset S^1\times S^{n-1}$$S^1\times*\subset S^1\times S^{n-1}$ to obtain an $n$$n$-sphere $\Sigma\subset C_{f_0}$$\Sigma\subset C_{f_0}$, and then we set $f_a:=f_0\# \Sigma$$f_a:=f_0\# \Sigma$.

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

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

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

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

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

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

4 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
• [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
• [Hirsch1959] M. W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR0119214 (22 #9980) Zbl 0118.18603
• [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
• [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.
• [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into $\Rr^{2n-k-1}$$\Rr^{2n-k-1}$, Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
• [Takase2006] M. Takase, Homology 3-spheres in codimension three, Internat. J. Math. 17 (2006), no.8, 869–885.

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

• [Vrabec1977] J. Vrabec, Knotting a $k$$k$-connected closed $\PL$$\PL$ $m$$m$-manifold in $E^{2m-k}$$E^{2m-k}$, Trans. Amer. Math. Soc. 233 (1977), 137–165. MR0645405 (58 #31097) Zbl 386.57013

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\in D^n$ on one component N be a closed connected $n$$n$-manifold. The Whitney invariant

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

is bijective if either $n\ge4$$n\ge4$ or $n=3$$n=3$ and CAT=PL [Haefliger&Hirsch1963], [Bausum1975], [Vrabec1977], cf. [Hudson1969]