3-manifolds in 6-space

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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.


For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$$\S$1, $\S$$\S$2].

2 Examples

For each integer $a$$a$ there is an embedding called the Hudson torus, $\Hud(a) \colon S^1\times S^2\to\Rr^6$$\Hud(a) \colon S^1\times S^2\to\Rr^6$, see [Skopenkov2016e, $\S$$\S$3].

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

Denote coordinates in $\Rr^6$$\Rr^6$ by $(x,y,z)=(x_1,x_2,y_1,y_2,z_1,z_2)$$(x,y,z)=(x_1,x_2,y_1,y_2,z_1,z_2)$. The Haefliger (higher-dimensional) trefoil knot $t$$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:

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

Example 2.2 (The Hopf embedding of $\Rr P^3$$\Rr P^3$ into $S^5$$S^5$). Represent $\Rr P^3=\{(x,y)\in\Cc^2\ |\ |x|^2+|y|^2=1\}/\pm1.$$\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$$h$ is an embedding. (The image of this embedding is given by the equations $b^2=4ac$$b^2=4ac$, $|a|^2+|b|^2+|c|^2=1$$|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$$f:\Rr P^3\to\Rr^6$ which is not isotopic to the composition of the Hopf embedding with the standard inclusion $S^5\subset\Rr^6$$S^5\subset\Rr^6$. (Such an embedding $f$$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$$\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$$R_i$ and $S_i$$S_i$ are real variables while $0$0 and $c_i$$c_i$ are constants. For various choices of $A_k$$A_k$ and $c_k$$c_k$ this system of equations defines embeddings of either $S^3$$S^3$, $S^1\times S^2$$S^1\times S^2$ or $\Rr P^3$$\Rr P^3$ into $\Rr^6$$\Rr^6$ [Bolsinov&Fomenko2004, Chapter 14].

3 Classification

The results of this subsection are proved in [Skopenkov2008] unless other references are given. Let $N$$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)$$W:E^6(N)\to H_1(N)$ is defined in [Skopenkov2016e]. For an abelian group $G$$G$ the divisibility of the identity element is zero and the divisibility of $x\in G-\{0\}$$x\in G-\{0\}$ is $\max\{d\in\Zz\ | \ \text{there is }x_1\in G:\ x=dx_1\}$$\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)}$$\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)$$a\in H_1(N)$ the Kreck invariant

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

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

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

Corollary 3.2 ([Haefliger1966], [Hausmann1972], [Takase2006]). (a) The Kreck invariant $\eta_0:E^6(N)\to\Zz$$\eta_0:E^6(N)\to\Zz$ is a 1--1 correspondence if $N$$N$ is $S^3$$S^3$ or an integral homology sphere. (For $N=S^3$$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$$H_2(N)=0$ (i.e. $N$$N$ is a rational homology sphere, e.g. $N=\Rr P^3$$N=\Rr P^3$), then $E^6(N)$$E^6(N)$ is in (non-canonical) 1-1 correspondence with $\Zz\times H_1(N)$$\Zz\times H_1(N)$.

(c) Isotopy classes of embeddings $S^1\times S^2\to\Rr^6$$S^1\times S^2\to\Rr^6$ with zero Whitney invariant are in 1-1 correspondence with $\Zz$$\Zz$, and for each integer $k\ne0$$k\ne0$ there are exactly $k$$k$ isotopy classes of embeddings $S^1\times S^2\to\Rr^6$$S^1\times S^2\to\Rr^6$ with the Whitney invariant $k$$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)$$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$$f:N\to\Rr^6$ is an embedding, $t$$t$ the generator of $E^6(S^3)\cong\Zz$$E^6(S^3)\cong\Zz$ and $kt$$kt$ is a connected sum of $k$$k$ copies of $t$$t$. Then $\eta_{W(f)}(f\#kt)\equiv\eta_{W(f)}(f)+k\mod d(W(f))$$\eta_{W(f)}(f\#kt)\equiv\eta_{W(f)}(f)+k\mod d(W(f))$.

E. g. for $N=\Rr P^3$$N=\Rr P^3$ the embedded connected sum action of $E^6(S^3)$$E^6(S^3)$ on $E^6(N)$$E^6(N)$ [Skopenkov2016c] is free while for $N=S^1\times S^2$$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$$f=\Hud(1):S^1\times S^2\to\Rr^6$ such that for each knot $g:S^3\to\Rr^6$$g:S^3\to\Rr^6$ the embedding $f\# g$$f\# g$ is isotopic to $f$$f$.

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

(We believe that this very corollary or the case $N=\Rr P^3$$N=\Rr P^3$ of Theorem 3.1 are as non-trivial as the general case of Theorem 3.1.)

4 The Kreck invariant

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

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

Definition 4.1. For a bundle isomorphism $\varphi$$\varphi$ denote

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

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

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

Definition 4.2. Take a small oriented disk $D^3_f\subset\Rr^6$$D^3_f\subset\Rr^6$ whose intersection with $fN$$fN$ consists of exactly one point of sign $+1$$+1$ and such that $\partial D^3_f\subset\partial C_f$$\partial D^3_f\subset\partial C_f$. A meridian of $f$$f$ is $\partial D^3_f$$\partial D^3_f$. A joint Seifert class for $f,f'$$f,f'$ and a bundle isomorphism $\varphi:\partial C_f\to\partial C_{f'}$$\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')$$W(f)=W(f')$ and $\varphi$$\varphi$ is a spin bundle isomorphism, then there is a joint Seifert class for $f,f'$$f,f'$ and $\varphi$$\varphi$ [Skopenkov2008, Agreement Lemma].

Denote by $PD:H^i(Q)\to H_{q-i}(Q,\partial Q)$$PD:H^i(Q)\to H_{q-i}(Q,\partial Q)$ and $PD:H_i(Q)\to H^{q-i}(Q,\partial Q)$$PD:H_i(Q)\to H^{q-i}(Q,\partial Q)$ Poincaré duality (in any oriented manifold $Q$$Q$).

Remark 4.3. The composition $H_4(C_f,\partial C_f)\to H_3(\partial C_f)\to H_3(N)$$H_4(C_f,\partial C_f)\to H_3(\partial C_f)\to H_3(N)$ of the boundary map $\partial$$\partial$ and the projection $\nu_f$$\nu_f$ is an isomorphism, cf. [Skopenkov2008, the Alexander Duality Lemma]. The inverse $A_f$$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)$$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$$f$ is the image $A_f[N]$$A_f[N]$ of the fundamental class $[N]$$[N]$.

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

For a joint Seifert class $Y\in H_4(M_\varphi)$$Y\in H_4(M_\varphi)$ for $f$$f$ and $f'$$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$$\Zz$ the zero-dimensional homology group of closed connected oriented manifols. The intersection products in 6-manifolds are omitted from the notation. Denote by $\sigma (X)$$\sigma (X)$ the signature of a 4-manifold $X$$X$. For a closed connected oriented 6-manifold $Q$$Q$ and $x\in H_4(Q)$$x\in H_4(Q)$ let the virtual signature of $(Q,x)$$(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]$$H_4(Q)\cong[Q,\Cc P^\infty]$, there is a closed connected oriented 4-submanifold $X\subset Q$$X\subset Q$ representing the class $x$$x$. Then $3\sigma(X)=PDp_1(X)=xPDp_1(Q)-x^3=3\sigma_x(Q)$$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$$f$ and $f'$$f'$ such that $W(f)=W(f')$$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))$$d:=d(W(f))$, $\rho_d$$\rho_d$ is the reduction modulo $d$$d$, $\varphi:\partial C_f\to\partial C_{f'}$$\varphi:\partial C_f\to\partial C_{f'}$ is a spin bundle isomorphism and $Y\in H_4(M)$$Y\in H_4(M)$ is a joint Seifert class for $f,f'$$f,f'$ and $\varphi$$\varphi$. Cf. [Ekholm2001, 4.1], [Zhubr2009].

We have $2Y\mod2=0=PDw_2(M_\varphi)$$2Y\mod2=0=PDw_2(M_\varphi)$, so any closed connected oriented 4-submanifold of $M_\varphi$$M_\varphi$ representing the class $2Y$$2Y$ is spin, hence by the Rokhlin Theorem $\sigma_{2Y}(M_\varphi)$$\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)$$a\in H_1(N)$ fix an embedding $f':N\to\Rr^6$$f':N\to\Rr^6$ such that $W(f')=a$$W(f')=a$ and define $\eta_a(f):=\eta(f,f')$$\eta_a(f):=\eta(f,f')$. (We write $\eta_a(f)$$\eta_a(f)$ not $\eta_{f'}(f)$$\eta_{f'}(f)$ for simplicity.)

The choice of the other orientation for $N$$N$ (resp. $\Rr^6$$\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)}$$W^{-1}(a)\to\Zz_{d(a)}$ (resp. replaces it with the bijection $W^{-1}(-a)\to\Zz_{d(a)}$$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$$\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$$f,f':N\to\Rr^6$ be two embeddings such that $W(f)=W(f')$$W(f)=W(f')$,
• $\varphi:\partial C_f\to\partial C_{f'}$$\varphi:\partial C_f\to\partial C_{f'}$ be a spin bundle isomorphism,
• $Y\subset M_\varphi$$Y\subset M_\varphi$ be a closed connected oriented 4-submanifold representing a joint Seifert class for $f,f',\varphi$$f,f',\varphi$ and
• $\overline p_1\in\Zz$$\overline p_1\in\Zz$, $\overline e\in H_2(Y)$$\overline e\in H_2(Y)$ be the Pontryagin number and Poincaré dual of the Euler classes of the normal bundle of $Y$$Y$ in $M_\varphi$$M_\varphi$.

Then

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