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$$== Introduction == <wikitex>; For notation and conventions see [[High_codimension_embeddings|high codimension embeddings]]. {{beginthm|Classification Theorem}}\label{th4} Let $N$ be a closed connected $n$-manifold. The Whitney invariant $$W:E^{2n}(N)\to\left\{\begin{array}{cc} H_1(N;\Zz_{(n-1)})& \mbox{either $n$ is odd or $N$ is orientable}\ \Zz\oplus\Zz_2^{s-1}&\mbox{$n$ is even, $N$ is non-orientable and } H_1(N,\Zz_2)\cong\Zz_2^s,\end{array}\right.$$ is bijective if either $n\ge4$ or $n=3$ and CAT=PL \cite{Haefliger&Hirsch1963}, \cite{Bausum1975}, \cite{Vrabec1977}, cf. \cite{Hudson1969} {{endthm}} [[Classification of smooth embeddings of 3-manifolds in the 6-space]] is more complicated. </wikitex> == Definition of the Whitney invariant (for either n odd or N orientable) == <wikitex>; Fix orientations on $\Rr^{2n}$ and, if $N$ is even, on $N$. Fix an embedding $f_0:N\to\Rr^{2n}$. For an embedding $f:N\to\Rr^{2n}$ the restrictions of $f$ and $f_0$ to $N_0$ are regular homotopic \cite{Hirsch1959}. Since $N_0$ has an $(n-1)$-dimensional spine, it follows that these restrictions are isotopic, cf. \cite{Haefliger&Hirsch1963}, 3.1.b, \cite{Takase2006}, Lemma 2.2. So we can make an isotopy of $f$ and assume that $f=f_0$ on $N_0$. Take a general position homotopy $F:B^n\times I\to\Rr^{2n}$ relative to $\partial B^n$ between the restrictions of $f$ and $g$ to $B^n$. Then $f\cap F:=(f|_{N-B^n})^{-1}F(B^n\times I)$ (i.e. `the intersection of this homotopy with $f(N-B^n)$') is a 1-manifold (possibly non-compact) without boundary. Define $W(f)$ to be the homology class of the closure of this 1-manifold: $$W(f):=[Cl(f\cap F)]\in H_1(N_0,\partial N_0;\Zz_{(n)})\cong H_1(N;\Zz_{(n)}).$$ The orientation on $f\cap F$ is defined for $N$ orientable as follows. (This orientation is defined for each $n$ but used only for odd $n$.) For each point $x\in f\cap F$ take a vector at $x$ tangent to $f\cap F$. Complete this vector to a positive base tangent to $N$. Since $n+2(n+1)>2\cdot2n$, by general position there is a unique point $y\in B^n\times I$ such that $Fy=fx$. The tangent vector at $x$ thus gives a tangent vector at $y$ to $B^n\times I$. Complete this vector to a positive base tangent to $B^n\times I$, where the orientation on $B^n$ comes from $N$. The union of the images of the constructed two bases is a base at $Fy=fx$ of $\Rr^{2n}$. If this base is positive, then call the initial vector of $f\cap F$ positive. Since a change of the orientation on $f\cap F$ forces a change of the orientation of the latter base of $\Rr^{2n}$, it follows that this condition indeed defines an orientation on $f\cap F$. {{beginthm|Remark}}\label{re5} *The Whitney invariant is well-defined, i.e. independent of the choice of $F$ and of the isotopy making $f=f_0$ outside $B^n$. This is so because the above definition is clearly equivalent to the following: $W(f)$ is the homology class of the algebraic sum of the top-dimensional simplices of the self-intersection set $\Sigma(H):=Cl\{x\in N\times I\ |\ \#H^{-1}Hx>1\}$ of a general position homotopy $H$ between $f$ and $f_0$. (For details and definition of the signs of the simplices see \cite{Hudson1969}, \S12, \cite{Vrabec1977}, p. 145, \cite{Skopenkov2006}, \S2, \cite{Skopenkov2010}.) It is for being well-defined that we need $\Zz_2$-coefficients when $n$ is even. *Clearly, $W(f_0)=0$. The definition of $W$ depends on the choice of $f_0$, but we write $W$ not $W_{f_0}$ for brevity. *Since a change of the orientation on $N$ forces a change of the orientation on $B^n$, the class $W(f)$ is independent of the choice of the orientation on $N$. For the reflection $\sigma:\Rr^{2n}\to\Rr^{2n}$ with respect to a hyperplane we have $W(\sigma\circ f)=-W(f)$ (because we may assume that $f=f_0=\sigma\circ f$ on $N_0$ and because a change of the orientation of $\Rr^{2n}$ forces a change of the orientation of $f\cap F$). *The above definition makes sense for each $n$, not only for $n\ge3$. *Clearly, $W(\Hud_n(a))$ is $a$ or $a\mod2$ for $n\ge2$ for the [[Hudson_tori|Hudson tori]]. *$W(f\#g)=W(f)$ for each embeddings $f:N\to\Rr^{2n}$ and $g:S^n\to\Rr^{2n}$. {{endthm}} </wikitex> == Construction of the embedding <wikitex> $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$) </wikitex>== <wikitex>; Represent $a$ by an embedding $a:S^1\to N$. Since any orientable bundle over $S^1$ is trivial, $\nu_{f_0}^{-1}a(S^1)\cong S^1\times S^{n-1}$. Identify $\nu_{f_0}^{-1}a(S^1)$ with $S^1\times S^{n-1}$. It remains to make an embedded surgery of $S^1\times*\subset S^1\times S^{n-1}$ to obtain an $n$-sphere $\Sigma\subset C_{f_0}$, and then we set $f_a:=f_0\# \Sigma$. Take a vector field on $S^1\times*$ normal to $S^1\times S^{n-1}$. Extend $S^1\times*$ along this vector field to a smooth map $\overline a:D^2\to S^{2n}$. Since n>4$ and $n+2<2n$, by general position we may assume that $\overline a$ is an embedding and $\overline a(Int D^2)$ misses $f_0(N)\cup S^1\times S^{n-1}$. Since $n-1>1$, we have $\pi_1(V_{2n-2,n-1})=0$. Hence the standard framing of $S^1\times*$ in $S^1\times S^{n-1}$ extends to an $(n-1)$-framing on $\overline a(D^2)$ in $\Rr^{2n}$. Thus $\overline a$ extends to an embedding $$\widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}$$ $$\mbox{Let}\qquad \Sigma:\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n.$$ Note that * This construction generalizes the construction of $\Hud_n(a)$ (from $\Hud_n(0)$). * Clearly, $W(f_a)$ is $a$ or $a\mod2$. Thus all isotopy classes of embedings $N\to\Rr^{2n}$ can be obtained (from a certain given embedding $f_0$) by the above construction. Hence unless $n=3$ and CAT=DIFF, the above construction defines an action $H_1(N;\Zz_{(n-1)})\to E^{2n}(N)$. </wikitex> == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]] {{Stub}}N$ be a closed connected $n$$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

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

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

Note that

• 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$$a\mod2$. Thus all isotopy classes of embedings $N\to\Rr^{2n}$$N\to\Rr^{2n}$ can be obtained (from a certain given embedding $f_0$$f_0$) by the above construction. Hence unless $n=3$$n=3$ and CAT=DIFF, 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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