3-manifolds in 6-space

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m (moved Examples of embeddings of 3-manifolds into the 6-space to Embeddings of 3-manifolds into 6-space: Aim to collect Arkadiy's 3 pages on E^6(N^3) into one page.)
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For notation and conventions see [[High_codimension_embeddings|high codimension embeddings]].
For notation and conventions see [[High_codimension_embeddings|high codimension embeddings]].
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== The Haefliger trefoil knot ==
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== Examples ==
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=== The Haefliger trefoil knot ===
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== The Hopf construction of an embedding <wikitex> $\Rr P^3\to S^5$ </wikitex> ==
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=== The Hopf construction of an embedding <wikitex> $\Rr P^3\to S^5$ </wikitex> ===
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== Algebraic embeddings from the theory of integrable systems ==
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=== Algebraic embeddings from the theory of integrable systems ===
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This defines embeddings of $S^3$, $S^1\times S^2$ or $\Rr P^3$ into $\Rr^6$.
This defines embeddings of $S^3$, $S^1\times S^2$ or $\Rr P^3$ into $\Rr^6$.
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== Invariants ==
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<wikitex>;
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For notation and conventions see [[High_codimension_embeddings|high codimension embeddings]]. For classification theorem involving the Kreck invariant see [[Classification_of_embeddings_of_3-manifolds_in_the_6-space|classification of embeddings of 3-manifolds in the 6-space]].
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Let $N$ be a closed connected orientable 3-manifold.
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An orientation-preserving diffeomorphism $\varphi:\partial C_f\to\partial C_{f'}$ such that $\nu_f=\nu_{f'}\varphi$ is simply called an $isomorphism$. For an isomorphism $\varphi$ denote
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$$M=M_\varphi:=C_f\cup_\varphi(-C_{f'}).$$
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An isomorphism $\varphi:\partial C_f\to\partial C_{f'}$ is called $spin$, if $\varphi$ over $N_0$ is defined by an isotopy between the restrictions of $f$ and $f'$ to $N_0$. A spin isomorphism exists because the restrictions to $N_0$ of $f$ and $f'$ are isotopic (see the definition of the Whitney invariant in [[Classification_just_below_the_stable_range|classification just below the stable range]]) and because $\pi_2(SO_3)=0$. If $\varphi$ is a spin isomorphism, then $M_\varphi$ is spin \cite{Skopenkov2008}, Spin Lemma.
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Denote by $\sigma (X)$ the signature of a 4-manifold $X$. Denote by $PD:H^i(Q)\to H_{q-i}(Q,\partial Q)$ and $PD:H_i(Q)\to H^{q-i}(Q,\partial Q)$ Poincar\'e duality (in any manifold $Q$). For $y\in H_4(M_\varphi)$ and a $k$-submanifold $C\subset M_\varphi$ (e.g. $C=C_f$ or $C=\partial C_f$) denote
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$$y\cap C:=PD[(PDy)|_C]\in H_{k-2}(C,\partial C).$$
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If $y$ is represented by a closed oriented 4-submanifold $Y\subset M_\varphi$ in general position to $C$, then $y\cap C$ is represented by $Y\cap C$.
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A $homology$ $Seifert$ $surface$ for $f$ is the image $A_f$ of the fundamental class $[N]$ under the composition $H_3(N)\to H^2(C_f)\to H_4(C_f,\partial C_f)$ of the Alexander and Poincar\'e duality isomorphisms. A $joint$ $homology$ $Seifert$ $surface$ for $f$ and $f'$ is a class $A\in H_4(M_\varphi)$ such that
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$$A\cap C_f=A_f\quad\text{and}\quad A\cap C_{f'}=A_{f'}.$$
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If $\varphi$ is a spin isomorphism and $W(f)=W(f')$, then there is a joint homology Seifert surface for $f$ and $f'$ \cite{Skopenkov2008}, Agreement Lemma.
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We identify with $\Zz$ the zero-dimensional homology groups and the $n$-dimensional cohomology groups of closed connected oriented $n$-manifolds. The intersection products in 6-manifolds are omitted from the notation. For a closed connected oriented 6-manifold $Q$ and $x\in H_4(Q)$ denote by
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$$\sigma_x(Q):=\frac{xPDp_1(Q)-x^3}3\in H_0(Q)=\Zz$$
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the virtual signature of $(Q,x)$.
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(Since $H_4(Q)\cong[Q,\Cc P^\infty]$, there is a closed connected oriented 4-submanifold $X\subset Q$ representing the class $x$. Then $3\sigma(X)=p_1(X)=xPDp_1(Q)-x^3=3\sigma_x(Q)$ by \cite{Hirzebruch1966}, end of 9.2 or else by \cite{Skopenkov2008}, Submanifold Lemma.)
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$The$ $Kreck$ $invariant$ of two embeddings $f$ and $f'$ such that $W(f)=W(f')$ is defined by
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$$\eta(f,f'):\equiv\frac{\sigma_{2A}(M_\varphi)}{16}=
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\frac{APDp_1(M_\varphi)-4A^3}{24}\mod d(W(f)),$$
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where $\varphi:\partial C_f\to\partial C_{f'}$ is a spin isomorphism and $A\in H_4(M)$ is a joint homology Seifert surface for $f$ and $f'$. Cf. \cite{Ekholm2001}, 4.1, \cite{Zhubr}. We have $2A\mod2=0=PDw_2(M_\varphi)$, so any closed connected oriented 4-submanifold of $M_\varphi$ representing the class $2A$ is spin, hence by the Rokhlin Theorem $\sigma_{2A}(M_\varphi)$ is indeed divisible by 16. The Kreck invariant is well-defined by \cite{Skopenkov2008}, Independence Lemma.
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For $a\in H_1(N)$ fix an embedding $f':N\to\Rr^6$ such that $W(f')=a$ and define $\eta_a(f):=\eta(f,f')$. (We write $\eta_a(f)$ not $\eta_{f'}(f)$ for simplicity.)
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The choice of the other orientation for $N$ (resp. $\Rr^6$) will in general give rise to different values for the Kreck invariant. But such a choice only permutes the bijection $W^{-1}(a)\to\Zz_{d(a)}$ (resp. replaces it with the bijection $W^{-1}(-a)\to\Zz_{d(a)}$).
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Let us present a formula for the Kreck invariant analogous to \cite{Guillou&Marin1986}, Remarks to the four articles of Rokhlin, II.2.7 and III.excercises.IV.3, \cite{Takase2004}, Corollary 6.5, \cite{Takase2006}, Proposition 4.1. This formula is useful when an embedding goes through $\Rr^5$ or is given by a system of equations (because we can obtain a `Seifert surface' by changing the equality to the inequality in one
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of the equations). See also
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[[http://www.ms.u-tokyo.ac.jp/~tetsuhir/mathematics/Integral-Formula.pdf|T. Moriyama, Integral formula for an extension of Haeliger's embedding invariant]].
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{{beginthm|The Kreck Invariant Lemma}}\label{th11}\cite{Skopenkov2008}
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Let $f,f':N\to\Rr^6$ be two embeddings such that $W(f)=W(f')$, $\varphi:\partial C_f\to\partial C_{f'}$ a spin isomorphism, $Y\subset M_\varphi$ a closed connected oriented 4-submanifold representing a joint homology Seifert surface and $\overline p_1\in\Zz$, $\overline e\in H_2(Y)$ are the Pontryagin number and Poincar\'e dual of the Euler classes of the normal bundle of $Y$ in $M_\varphi$. Then
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$$\frac{\sigma_{2[Y]}(M_\varphi)}{16}=\frac{\sigma(Y)-\overline p_1}8=
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\frac{\sigma(Y)-\overline e\cap\overline e}8.$$
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{{endthm}}
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</wikitex>
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== References ==
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{{#RefList:}}
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[[Category:Manifolds]]
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[[Category:Embeddings of manifolds]]
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{{Stub}}
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== References ==
== References ==

Revision as of 09:09, 19 February 2010

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

For notation and conventions see high codimension embeddings.

Contents

1 Examples

1.1 The Haefliger trefoil knot

Let us construct a smooth embedding t:S^3\to\Rr^6 (which is a generator of E^6(S^3)\cong\Zz) [Haefliger1962], 4.1. A miraculous property of this embedding is that it is not smoothly isotopic to the standard embedding, but is piecewise smoothly isotopic to the standard embedding.

Denote coordinates in \Rr^6 by (x,y,z)=(x_1,x_2,y_1,y_2,z_1,z_2). The higher-dimensional trefoil knot t is obtained by joining with two tubes the higher-dimensional Borromean rings, i.e. the three spheres given by the following three systems of equations:

\displaystyle \left\{\begin{array}{c} x=0\\ |y|^2+2|z|^2=1\end{array}\right., \qquad \left\{\begin{array}{c} y=0\\ |z|^2+2|x|^2=1\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} z=0\\ |x|^2+2|y|^2=1 \end{array}\right..

See Figures 3.5 and 3.6 of [Skopenkov2006].

1.2 The Hopf construction of an embedding \Rr P^3\to S^5

Represent \Rr P^3=\{(x,y)\in\Cc^2\ |\ |x|^2+|y|^2=1\}/\pm1. Define

\displaystyle Ho:\Rr P^3\to S^5\subset\Cc^3\quad\text{by}\quad Ho[(x,y)]=(x^2,2xy,y^2).

It is easy to check that Ho is an embedding. (The image of this embedding is given by the equations b^2=4ac, |a|^2+|b|^2+|c|^2=1.)

It would be interesting to obtain an explicit construction of an embedding f:\Rr P^3\to S^6 which is not isotopic to the composition of the Hopf embedding with the standard inclusion S^5\subset\Rr^6. (Such an embedding f is unique up to PL isotopy by classification just below the stable range.)

1.3 Algebraic embeddings from the theory of integrable systems

Some 3-manifolds appear in the theory of integrable systems together with their embeddings into \Rr^6 (given by a system of algebraic equations) [Bolsinov&Fomenko2004], Chapter 14. E.g. the following system of equations corresponds to the Euler integrability case:

\displaystyle R_1^2+R_2^2+R_3^2=c_1,\quad R_1S_1+R_2S_2+R_3S_3=c_2,\quad \frac{S_1^2}{A_1}+\frac{S_2^2}{A_2}+\frac{S_3^2}{A_3}=c_3,

where R_i and S_i are variables while A_1<A_2<A_3 and c_i are constants. This defines embeddings of S^3, S^1\times S^2 or \Rr P^3 into \Rr^6.

2 Invariants

For notation and conventions see high codimension embeddings. For classification theorem involving the Kreck invariant see classification of embeddings of 3-manifolds in the 6-space. Let N be a closed connected orientable 3-manifold.

An orientation-preserving diffeomorphism \varphi:\partial C_f\to\partial C_{f'} such that \nu_f=\nu_{f'}\varphi is simply called an isomorphism. For an isomorphism \varphi denote

\displaystyle M=M_\varphi:=C_f\cup_\varphi(-C_{f'}).

An isomorphism \varphi:\partial C_f\to\partial C_{f'} is called spin, if \varphi over N_0 is defined by an isotopy between the restrictions of f and f' to N_0. A spin isomorphism exists because the restrictions to N_0 of f and f' are isotopic (see the definition of the Whitney invariant in classification just below the stable range) and because \pi_2(SO_3)=0. If \varphi is a spin isomorphism, then M_\varphi is spin [Skopenkov2008], Spin Lemma.

Denote by \sigma (X) the signature of a 4-manifold X. Denote by PD:H^i(Q)\to H_{q-i}(Q,\partial Q) and PD:H_i(Q)\to H^{q-i}(Q,\partial Q) Poincar\'e duality (in any manifold Q). For y\in H_4(M_\varphi) and a k-submanifold C\subset M_\varphi (e.g. C=C_f or C=\partial C_f) denote

\displaystyle y\cap C:=PD[(PDy)|_C]\in H_{k-2}(C,\partial C).

If y is represented by a closed oriented 4-submanifold Y\subset M_\varphi in general position to C, then y\cap C is represented by Y\cap C.

A homology Seifert surface for f is the image A_f of the fundamental class [N] under the composition H_3(N)\to H^2(C_f)\to H_4(C_f,\partial C_f) of the Alexander and Poincar\'e duality isomorphisms. A joint homology Seifert surface for f and f' is a class A\in H_4(M_\varphi) such that

\displaystyle A\cap C_f=A_f\quad\text{and}\quad A\cap C_{f'}=A_{f'}.

If \varphi is a spin isomorphism and W(f)=W(f'), then there is a joint homology Seifert surface for f and f' [Skopenkov2008], Agreement Lemma.

We identify with \Zz the zero-dimensional homology groups and the n-dimensional cohomology groups of closed connected oriented n-manifolds. The intersection products in 6-manifolds are omitted from the notation. For a closed connected oriented 6-manifold Q and x\in H_4(Q) denote by

\displaystyle \sigma_x(Q):=\frac{xPDp_1(Q)-x^3}3\in H_0(Q)=\Zz

the virtual signature of (Q,x). (Since H_4(Q)\cong[Q,\Cc P^\infty], there is a closed connected oriented 4-submanifold X\subset Q representing the class x. Then 3\sigma(X)=p_1(X)=xPDp_1(Q)-x^3=3\sigma_x(Q) by [Hirzebruch1966], end of 9.2 or else by [Skopenkov2008], Submanifold Lemma.)

The Kreck invariant of two embeddings f and f' such that W(f)=W(f') is defined by

\displaystyle \eta(f,f'):\equiv\frac{\sigma_{2A}(M_\varphi)}{16}= \frac{APDp_1(M_\varphi)-4A^3}{24}\mod d(W(f)),

where \varphi:\partial C_f\to\partial C_{f'} is a spin isomorphism and A\in H_4(M) is a joint homology Seifert surface for f and f'. Cf. [Ekholm2001], 4.1, [Zhubr]. We have 2A\mod2=0=PDw_2(M_\varphi), so any closed connected oriented 4-submanifold of M_\varphi representing the class 2A is spin, hence by the Rokhlin Theorem \sigma_{2A}(M_\varphi) is indeed divisible by 16. The Kreck invariant is well-defined by [Skopenkov2008], Independence Lemma.

For a\in H_1(N) fix an embedding f':N\to\Rr^6 such that W(f')=a and define \eta_a(f):=\eta(f,f'). (We write \eta_a(f) not \eta_{f'}(f) for simplicity.)

The choice of the other orientation for N (resp. \Rr^6) will in general give rise to different values for the Kreck invariant. But such a choice only permutes the bijection W^{-1}(a)\to\Zz_{d(a)} (resp. replaces it with the bijection W^{-1}(-a)\to\Zz_{d(a)}).

Let us present a formula for the Kreck invariant analogous to [Guillou&Marin1986], Remarks to the four articles of Rokhlin, II.2.7 and III.excercises.IV.3, [Takase2004], Corollary 6.5, [Takase2006], Proposition 4.1. This formula is useful when an embedding goes through \Rr^5 or is given by a system of equations (because we can obtain a `Seifert surface' by changing the equality to the inequality in one of the equations). See also [Moriyama, Integral formula for an extension of Haeliger's embedding invariant].

The Kreck Invariant Lemma 2.1.[Skopenkov2008] Let f,f':N\to\Rr^6 be two embeddings such that W(f)=W(f'), \varphi:\partial C_f\to\partial C_{f'} a spin isomorphism, Y\subset M_\varphi a closed connected oriented 4-submanifold representing a joint homology Seifert surface and \overline p_1\in\Zz, \overline e\in H_2(Y) are the Pontryagin number and Poincar\'e dual of the Euler classes of the normal bundle of Y in M_\varphi. Then

\displaystyle \frac{\sigma_{2[Y]}(M_\varphi)}{16}=\frac{\sigma(Y)-\overline p_1}8= \frac{\sigma(Y)-\overline e\cap\overline e}8.

3 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.


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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