3-manifolds in 6-space

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This page is being refereed under the supervision of the editorial board. Hence the page may not be edited at present. As always, the discussion page remains open for observations and comments.

Contents

1 Introduction

This page is intended not only for specialists in embeddings, but also for mathematicians from other areas who want to apply or to learn the theory of embeddings.

The classification of 3-manifolds in 6-space is of course a particular case of the classification of n-manifolds in 2n-space which is discussed in [Skopenkov2016e]. In this page we recall the general results as they apply when n = 3 and we discuss examples and invariants peculiar to the case n=3.

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, \S1, \S2].

2 Examples

For each integer a there is an embedding called the Hudson torus, \Hud(a) \colon S^1\times S^2\to\Rr^6, see [Skopenkov2016e, \S3].

Example 2.1 (The Haefliger trefoil knot). There is a smooth embedding t:S^3\to\Rr^6 with a surprising property that it is not smoothly isotopic to the standard embedding [Haefliger1962], but is piecewise smoothly isotopic to the standard embedding (by the Zeeman Unknotting Spheres Theorem 2.3 [Skopenkov2016c]). (This embedding is a generator of E^6_D(S^3)\cong\Zz) [Haefliger1962, 4.1].

Denote coordinates in \Rr^6 by (x,y,z)=(x_1,x_2,y_1,y_2,z_1,z_2). The Haefliger (higher-dimensional) trefoil knot t is obtained by joining with two tubes the higher-dimensional Borromean rings [Skopenkov2006, Figures 3.5 and 3.6], 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 motivating examples of links [Skopenkov2016h] and a higher-dimensional generalization [Skopenkov2016k].

Example 2.2 (The Hopf embedding of \Rr P^3 into S^5). Represent \Rr P^3=\{(x,y)\in\Cc^2\ |\ |x|^2+|y|^2=1\}/\pm1. Define

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

It is easy to check that h 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\Rr^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 results just below the stable range, see [Skopenkov2016e, Theorem 2.1].)

Example 2.3 (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 [Bolsinov&Fomenko2004, Chapter 14]:

\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 real variables while 0<A_1<A_2<A_3 and c_i are constants. For various choices of A_k and c_k this system of equations defines embeddings of either S^3, S^1\times S^2 or \Rr P^3 into \Rr^6 [Bolsinov&Fomenko2004, Chapter 14].

3 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. For a classification in the PL category see [Skopenkov2016e, Theorem 2.1].

For the next theorem, the Whitney invariant W:E^6(N)\to H_1(N) is defined in [Skopenkov2016e]. For an abelian group G the divisibility of the identity element 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\}. The Kreck invariant \eta_a:W^{-1}(u)\to\Zz_{d(a)} is defined in Section 4 below.

Theorem 3.1. 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).

Cf. a higher-dimensional generalization [Skopenkov2016e].

All isotopy classes of embeddings N\to\Rr^6 can be constructed from a certain given embedding using unlinked and linked embedded connected sum with embeddings S^3\to\Rr^6 [Skopenkov2016c], [Skopenkov2016e].

Corollary 3.2 ([Haefliger1966], [Hausmann1972], [Takase2006]). (a) 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. (For N=S^3 the Kreck invariant is also a group isomorphism; this follows not from Theorem 3.1 but from [Haefliger1966].)

(b) 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).

(c) Isotopy classes of 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 isotopy classes of embeddings S^1\times S^2\to\Rr^6 with the Whitney invariant k, cf. Corollary 3.4 below.

(d) 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

\displaystyle  |W^{-1}(a_1\oplus a_2)|=\begin{cases} \infty & d(a_1)=d(a_2)=0\\  \gcd(d(a_1),d(a_2)) &\text{otherwise} \end{cases}.

Addendum 3.3. Let f:N\to\Rr^6 is an embedding, t 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 embedded connected sum action of E^6(S^3) on E^6(N) [Skopenkov2016c] is free while for N=S^1\times S^2 we have the following corollary.

Corollary 3.4. (a) The Hudson torus Hud(1) 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.

(b) 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 3.1 are as non-trivial as the general case of Theorem 3.1.)

See also [Avvakumov2016].

4 The Kreck invariant

We work in the smooth category and use notation and conventions [Skopenkov2016c, \S3]. Let N be a closed connected orientable 3-manifold and f,f':N\to\Rr^6 embeddings. Fix orientations on N and on \Rr^6.

An orientation-preserving diffeomorphism \varphi:\partial C_f\to\partial C_{f'} such that \nu_f=\nu_{f'}\varphi is called a bundle isomorphism. (By the Smale Theorem [Smale1959] this is equivalent to \varphi being isotopic to the restriction of a vector bundle isomorphism to the spherical bundle.)

Definition 4.1. For a bundle isomorphism \varphi denote

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

A bundle isomorphism \varphi:\partial C_f\to\partial C_{f'} is called spin, if M_\varphi is spin.

A spin bundle isomorphism exists. Indeed, the restrictions to N_0 of f and f' are isotopic (this is proved in definition of the Whitney invariant [Skopenkov2016e]). Define \varphi over N_0 by an isotopy between the restrictions to N_0 of f and f'. Since \pi_2(SO_3)=0, \varphi extends to N. Then M_\varphi is spin. Cf. [Skopenkov2008, Spin Lemma].

Definition 4.2. Take a small oriented disk D^3_f\subset\Rr^6 whose intersection with fN consists of exactly one point of sign +1 and such that \partial D^3_f\subset\partial C_f. A meridian of f is \partial D^3_f. A joint Seifert class for f,f' and a bundle isomorphism \varphi:\partial C_f\to\partial C_{f'} is a class

\displaystyle  Y\in H_5(M_\varphi)\quad\text{such that}\quad Y\cap [\partial D^3_f]=1.

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

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) Poincaré duality (in any oriented manifold Q).

Remark 4.3. 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 is an isomorphism, cf. [Skopenkov2008, the Alexander Duality Lemma]. The inverse A_f to this composition is homology Alexander Duality isomorphism; it equals to the composition H_3(N)\to H^2(C_f)\to H_4(C_f,\partial C_f) of the cohomology Alexander and Poincaré duality isomorphisms.

A homology Seifert surface for f is the image A_f[N] of the fundamental class [N].

For Y\in H_4(M_\varphi) denote Y\cap C_f:=PD[(PDY)|_{C_f}]\in H_4(C_f,\partial C_f). If Y is represented by a closed oriented 4-submanifold Q\subset M_\varphi in general position to C_f, then Y\cap C_f is represented by Q\cap C_f.

For a joint Seifert class Y\in H_4(M_\varphi) for f and f' we have

\displaystyle  Y\cap C_f=A_f[N]\quad\text{and}\quad Y\cap C_{f'}=A_{f'}[N].

This property provides an equivalent definition of a joint Seifert class which explains the name and which was used in [Skopenkov2008] together with the name `joint homology Seifert surface'.

Identify with \Zz the zero-dimensional homology group of closed connected oriented manifols. The intersection products in 6-manifolds are omitted from the notation. Denote by \sigma (X) the signature of a 4-manifold X. For a closed connected oriented 6-manifold Q and x\in H_4(Q) let the virtual signature of (Q,x) be

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

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)=PDp_1(X)=xPDp_1(Q)-x^3=3\sigma_x(Q) by [Hirzebruch1966, end of 9.2] or else by [Skopenkov2008, Submanifold Lemma].

Definition 4.4. The Kreck invariant of two embeddings f and f' such that W(f)=W(f') is defined by

\displaystyle \eta(f,f'):=\rho_d\frac{\sigma_{2Y}(M_\varphi)}{16}=\rho_d\frac{PDp_1(M_\varphi)Y-4Y^3}{24}\in\Zz_d,

where d:=d(W(f)), \rho_d is the reduction modulo d, \varphi:\partial C_f\to\partial C_{f'} is a spin bundle isomorphism and Y\in H_4(M) is a joint Seifert class for f,f' and \varphi. Cf. [Ekholm2001, 4.1], [Zhubr2009].

We have 2Y\mod2=0=PDw_2(M_\varphi), so any closed connected oriented 4-submanifold of M_\varphi representing the class 2Y is spin, hence by the Rokhlin Theorem \sigma_{2Y}(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], [Moriyama2008].

The Kreck Invariant Lemma 4.5 ([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'} be a spin bundle isomorphism,
  • Y\subset M_\varphi be a closed connected oriented 4-submanifold representing a joint Seifert class for f,f',\varphi and
  • \overline p_1\in\Zz, \overline e\in H_2(Y) be the Pontryagin number and Poincaré 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.

5 References

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