3-manifolds in 6-space

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

For notation and conventions throughout this page 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_D(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].

Analogously for k>1 one constructs a smooth embedding t:S^{2k-1}\to\Rr^{3k} (which is a generator of E_D^{3k}(S^3)\cong\Zz_{(k)}) that is not smoothly isotopic to the standard embedding, but is piecewise smoothly isotopic to it [Haefliger1962].

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 Classification

The results of this subsection are proved in [Skopenkov2008] unless other references are given. Let N be a closed connected orientable 3-manifold. We work in the smooth category. A classification in the PL category.

Theorem 2.1. The Whitney invariant

\displaystyle W:E^6(N)\to H_1(N)

is surjective. For each a\in H_1(N) the Kreck invariant

\displaystyle \eta_a:W^{-1}(u)\to\Zz_{d(a)}

is bijective, where d(a) is the divisibility of the projection of a to the free part of H_1(N).

Recall that for an abelian group G 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\}.

Cf. higher-dimensional generalization.

All isotopy classes of embeddings N\to\Rr^6 can be constructed (from a certain given embedding) using connected sum with embeddings S^3\to\Rr^6, see the construction in Embeddings just below the stable range and in High codimension embeddings.

Corollary 2.2.

  1. The Kreck invariant \eta_0:E^6(N)\to\Zz is a 1--1 correspondence if N is S^3 or an integral homology sphere [Haefliger1966], [Hausmann1972], [Takase2006]. (For N=S^3 the Kreck invariant is also a group isomorphism; this follows not from Theorem 2.1 but from [Haefliger1966].)
  2. If H_2(N)=0 (i.e. N is a rational homology sphere, e.g. N=\Rr P^3), then E^6(N) is in (non-canonical) 1-1 correspondence with \Zz\times H_1(N).
  3. Embeddings S^1\times S^2\to\Rr^6 with zero Whitney invariant are in 1-1 correspondence with \Zz, and for each integer k\ne0 there are exactly k distinct (i.e. non-isotopic) embeddings S^1\times S^2\to\Rr^6 with the Whitney invariant k, cf. Corollary 2.3 below.
  4. The Whitney invariant W:E^6(N_1\# N_2)\to H_1(N_1\# N_2)\cong H_1(N_1)\oplus H_1(N_2) is surjective and \# W^{-1}(a_1\oplus a_2)=\#\Zz_{GCD(d(a_1),d(a_2))}.

If f:N\to\Rr^6 is an embedding, t is the generator of E^6(S^3)\cong\Zz and kt is a connected sum of k copies of t, then \eta_{W(f)}(f\#kt)\equiv\eta_{W(f)}(f)+k\mod d(W(f)). E. g. for N=\Rr P^3 the action \#:E^6(S^3)\to E^6(N) is free while for N=S^1\times S^2 we have the following corollary.

Corollary 2.3.

  • There is an embedding f=\Hud(1):S^1\times S^2\to\Rr^6 such that for each knot g:S^3\to\Rr^6 the embedding f\# g is isotopic to f.
  • For each embedding f:N\to\Rr^6 such that f(N)\subset\Rr^5 (e.g. for the standard embedding f:S^1\times S^2\to\Rr^6) and each non-trivial knot g:S^3\to\Rr^6 the embedding f\# g is not isotopic to f.

(We believe that this very corollary or the case N=\Rr P^3 of Theorem 2.1 are as non-trivial as the general case of Theorem 2.1.)

3 The Kreck invariant

Let N be a closed connected orientable 3-manifold. We work in the smooth category. Fix orientations on N and on S^7.

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 Whitney invariant) 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. (This composition is an inverse to the composition H_4(C_f,\partial C_f)\to H_3(\partial C_f)\to H_3(N) of the boundary map \partial and the projection \nu_f, cf. [Skopenkov2008], the Alexander Duality Lemma; this justifies the name `homology Seifert surface'.)

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.)

Definition 3.1. 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, [Zhubr2009].

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 T. Moriyama, Integral formula for an extension of Haeliger's embedding invariant.

The Kreck Invariant Lemma 3.2.[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.

4 References

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

  • [Zhubr2009] A. V. Zhubr, Exotic invariant for 6-manifolds: a direct construction, Algebra i Analiz, 21:3 (2009), 145–164. English translation: St.Petersburg Math. J., 21:3 (2009).

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

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