Embeddings just below the stable range: classification

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\cite{Hudson1969}
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== Introduction ==
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<wikitex>;
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== References ==
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For notation and conventions see [[High_codimension_embeddings|high codimension embeddings]].
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{{beginthm|Classification Theorem}}\label{th4}
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Let $N$ be a closed connected $n$-manifold. The Whitney invariant
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$$W:E^{2n}(N)\to \cases H_1(N;\Zz_{(n-1)})&
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\text{either $n$ is odd or $N$ is orientable}\\
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\Zz\oplus\Zz_2^{s-1}&\text{$n$ is even, $N$ is non-orientable and }
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H_1(N,\zZ_2)\cong\Zz_2^s,\endcases$$
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is bijective if either $n\ge4$ or $n=3$ and CAT=PL \cite{HaefligerHirsch1963}, \cite{Bausum75}, \cite{Vrabec77}, cf. \cite{Hudson1969}
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{{endthm}}
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[[Classification of smooth embeddings of 3-manifolds in the 6-space]] is more complicated.
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</wikitex>
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== Definition of the Whitney invariant (for either n odd or N orientable) ==
<wikitex>;
<wikitex>;
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{{#RefList:}}
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Fix orientations on $\R^{2n}$ and, if $N$ is even, on $N$. Fix an embedding $f_0:N\to\Rr^{2n}$. For an embedding $f:N\to\Rr^{2n}$
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the restrictions of $f$ and $f_0$ to $N_0$ are regular homotopic \cite{Hirsch1960}. Since $N_0$ has an $(n-1)$-dimensional spine, it follows that these restrictions are isotopic, cf. \cite{HaefligerHirsch1963}, 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
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$\partial B^n$ 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:
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$$W(f):=[\Cl(f\cap F)]\in H_1(N_0,\partial N_0;\Zz_{(n)})\cong H_1(N;\Zz_{(n)}).$$
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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 $f\cap F$.
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{{beginthm|Remark}}\label{re5}
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*The Whitney invariant is well-defined, i.e. independent of the choice of $F$ and of the isotopy making $f=f_0$ outside $B^n$. This is so because the above definition is clearly equivalent to the following: $W(f)$ is the homology class of the algebraic sum of the top-dimensional simplices of the self-intersection set $\Sigma(H):=Cl\{x\in N\times I\ |\ \#H^{-1}Hx>1\}$ of a general position homotopy $H$ between $f$ and $f_0$.
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(For details and definition of the signs of the simplices see \cite{Hudson1969}, \S12, \cite{Vrabec1977}, p. 145, \cite{Skopenkov2006}, \S2, \cite{Skopenkov2010}.)
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It is for being well-defined that we need $\Zz_2$-coefficients when $n$ is even.
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*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.
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*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 $f\cap F$).
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*The above definition makes sense for each $n$, not only for $n\ge3$.
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*Clearly, $W(\Hud_n(a))$ is $a$ or $a\mod2$ for $n\ge2$ for the [[Hudson_tori|Hudson tori]].
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*$W(f\#g)=W(f)$ for each embeddings $f:N\to\Rr^{2n}$ and $g:S^n\to\Rr^{2n}$.
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{{endthm}}
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</wikitex>
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== Construction of the embedding <wikitex>; $f_a:N\to\R^{2n}$ from an embedding
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$f_0:N\to\R^{2n}$ and $a\in H_1(N)$ (for orientable $N$ and $n\ge3$) </wikitex>==
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<wikitex>;
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Represent $a$ by an embedding $a:S^1\to N$. Since any orientable bundle over $S^1$ is trivial, $\nu_{f_0}^{-1}a(S^1)\cong S^1\times S^{n-1}$. Identify $\nu_{f_0}^{-1}a(S^1)$ with $S^1\times S^{n-1}$. It remains to make an embedded surgery of
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$S^1\times*\subset S^1\times S^{n-1}$ to obtain an $n$-sphere $\Sigma\subset C_{f_0}$, and then we set $f_a:=f_0\# \Sigma$.
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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
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$\overline a:D^2\to S^{2n}$. 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 $\pi_1(V_{2n-2,n-1})=0$.
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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 $\Rr^{2n}$.
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Thus $\overline a$ extends to an embedding
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$$\widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad
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\widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}$$
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$$\text{Let}\qquad
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\Sigma:\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times\Int D^{n-1})
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\bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})}
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\widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n.$$
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Note that
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* This construction generalizes the construction of $\Hud_n(a)$ (from $\Hud_n(0)$).
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* Clearly, $W(f_a)$ is $a$ or $a\mod2$. Thus all isotopy classes of embedings $N\to\Rr^{2n}$ can be obtained (from a certain given embedding $f_0$) by the above construction. Hence unless $n=3$ and CAT=DIFF, the above construction defines an action
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$H_1(N;\Zz_{(n-1)})\to E^{2n}(N)$.
</wikitex>
</wikitex>
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== References ==
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{{#RefList:}}
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[[Category:Manifolds]]
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{{Stub}}

Revision as of 15:39, 14 February 2010

This page has been accepted for publication in the Bulletin of the Manifold Atlas.

Contents

1 Introduction

For notation and conventions see high codimension embeddings.

Classification Theorem 1.1. Let N be a closed connected n-manifold. The Whitney invariant 

Tex syntax error

is bijective if either n\ge4 or n=3 and CAT=PL [HaefligerHirsch1963], [Bausum75], [Vrabec77], 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 \R^{2n} and, if N is even, on N. Fix an embedding f_0:N\to\Rr^{2n}. For an embedding f:N\to\Rr^{2n} the restrictions of f and f_0 to N_0 are regular homotopic [Hirsch1960]. Since N_0 has an (n-1)-dimensional spine, it follows that these restrictions are isotopic, cf. [HaefligerHirsch1963], 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: 

Tex syntax error

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 2.1.

  • The Whitney invariant is well-defined, i.e. independent of the choice of F and of the isotopy making f=f_0 outside B^n. This is so because the above definition is clearly equivalent to the following: W(f) is the homology class of the algebraic sum of the top-dimensional simplices of the self-intersection set \Sigma(H):=Cl\{x\in N\times I\ |\ \#H^{-1}Hx>1\} of a general position homotopy H between f and f_0.

(For details and definition of the signs of the simplices see [Hudson1969], \S12, [Vrabec1977], p. 145, [Skopenkov2006], \S2, [Skopenkov2010].) It is for being well-defined that we need \Zz_2-coefficients when n is even.

  • Clearly, W(f_0)=0. The definition of W depends on the choice of f_0, but we write W not W_{f_0} for brevity.
  • Since a change of the orientation on N forces a change of the orientation on B^n, the class W(f) is independent of the choice of the orientation on N. For the reflection \sigma:\Rr^{2n}\to\Rr^{2n} with respect to a hyperplane we have W(\sigma\circ f)=-W(f) (because we may assume that f=f_0=\sigma\circ f on N_0 and because a change of the orientation of \Rr^{2n} forces a change of the orientation of f\cap F).
  • The above definition makes sense for each n, not only for n\ge3.
  • Clearly, W(\Hud_n(a)) is a or a\mod2 for n\ge2 for the Hudson tori.
  • W(f\#g)=W(f) for each embeddings f:N\to\Rr^{2n} and g:S^n\to\Rr^{2n}.


Construction of the embedding ; Tex syntax errorf a:N\to\R^{2n} from an embedding Tex syntax errorf 0:N\to\R^{2n} and a\in H 1(N) (for orientable N and n\ge3) 3 Construction of the embedding ; 
Tex syntax error
 from an embedding 
Tex syntax error
 and a\in H_1(N) (for orientable N and n\ge3)

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

Take a vector field on S^1\times* normal to S^1\times S^{n-1}. Extend S^1\times* along this vector field to a smooth map

\overline a:D^2\to S^{2n}. Since 2n>4 and n+2<2n, by general position we may assume that \overline a is an embedding and 
Tex syntax error
 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}
Tex syntax error

Note that

  • This construction generalizes the construction of \Hud_n(a) (from \Hud_n(0)).
  • Clearly, W(f_a) is a or a\mod2. Thus all isotopy classes of embedings N\to\Rr^{2n} can be obtained (from a certain given embedding f_0) by the above construction. Hence unless n=3 and CAT=DIFF, the above construction defines an action

H_1(N;\Zz_{(n-1)})\to E^{2n}(N).

4 References

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

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

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