3-manifolds in 6-space

Latest revision as of 13:02, 25 June 2019

1 Introduction

Most of 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.

Basic results on embeddings of closed connected 3-manifolds in 6-space are particular cases of results on embeddings of $For notation and conventions throughout this page see [[High_codimension_embeddings|high codimension embeddings]]. == Examples == === The Haefliger trefoil knot === <wikitex>; Let us construct a smooth embedding $t:S^3\to\Rr^6$ (which is a generator of $E^6_D(S^3)\cong\Zz$) \cite{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: $$\left\{\begin{array}{c} x=0\ |y|^2+2|z|^2=1\end{array}\right., \qquad \left\{\begin{array}{c} y=0\ |z|^2+2|x|^2=1\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} z=0\ |x|^2+2|y|^2=1 \end{array}\right..$$ See Figures 3.5 and 3.6 of \cite{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 \cite{Haefliger1962}. </wikitex> === The Hopf construction of an embedding <wikitex> $\Rr P^3\to S^5$ </wikitex> === <wikitex>; Represent $\Rr P^3=\{(x,y)\in\Cc^2\ |\ |x|^2+|y|^2=1\}/\pm1.$ Define $$Ho:\Rr P^3\to S^5\subset\Cc^3\quad\text{by}\quad Ho[(x,y)]=(x^2,2xy,y^2).$$ 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 [[Embeddings_just_below_the_stable_range:_classification#Classification|classification just below the stable range]].) </wikitex> === Algebraic embeddings from the theory of integrable systems === <wikitex>; Some 3-manifolds appear in the theory of integrable systems together with their embeddings into $\Rr^6$ (given by a system of algebraic equations) \cite{Bolsinov&Fomenko2004}, Chapter 14. E.g. the following system of equations corresponds to the Euler integrability case: $$R_1^2+R_2^2+R_3^2=c_1,\quad R_1S_1+R_2S_2+R_3S_3=c_2,\quad \frac{S_1^2}{A_1}+\frac{S_2^2}{A_2}+\frac{S_3^2}{A_3}=c_3,$$ 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 $\Rr^6$. </wikitex> == Classification == <wikitex>; The results of this subsection are proved in \cite{Skopenkov2008} unless other references are given. Let $N$ be a closed connected orientable 3-manifold. We work in the smooth category. [[Embeddings just below the stable range: classification#Classification|A classification in the PL category]]. {{beginthm|Theorem}}\label{th7} [[Embeddings_just_below_the_stable_range:_classification#The Whitney invariant (for either n even or N orientable)|The Whitney invariant]] $$W:E^6(N)\to H_1(N)$$ is surjective. For each $a\in H_1(N)$ [[Embeddings_of_3-manifolds_in_6-space#The Kreck invariant|the Kreck invariant]] $$\eta_a:W^{-1}(u)\to\Zz_{d(a)}$$ is bijective, where $d(a)$ is the divisibility of the projection of $a$ to the free part of $H_1(N)$. {{endthm}} 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. [[Embeddings just below the stable range: classification#A generalization to highly-connected manifolds|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 [[Classification_just_below_the _stable_range#An action of the first homology group on embeddings|Embeddings just below the stable range]] and in [[High codimension embeddings#Embedded connected sum|High codimension embeddings]]. {{beginthm|Corollary}}\label{co8} #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 \cite{Haefliger1966}, \cite{Hausmann1972}, \cite{Takase2006}. (For $N=S^3$ the Kreck invariant is also a group isomorphism; this follows not from Theorem \ref{th7} but from \cite{Haefliger1966}.) #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 $\Zz\times H_1(N)$. #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 \ref{co10} below. #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 $\# W^{-1}(a_1\oplus a_2)=\#\Zz_{GCD(d(a_1),d(a_2))}$. {{endthm}} 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 [[High codimension embeddings: classification#Embedded connected sum|action]] $\#:E^6(S^3)\to E^6(N)$ is free while for $N=S^1\times S^2$ we have the following corollary. {{beginthm|Corollary}}\label{co10} *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 $f$. {{endthm}} (We believe that this very corollary or the case $N=\Rr P^3$ of Theorem \ref{th7} are as non-trivial as the general case of Theorem \ref{th7}.) </wikitex> == The Kreck invariant == <wikitex>; 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 $$M=M_\varphi:=C_f\cup_\varphi(-C_{f'}).$$ 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 [[Embeddings_just_below_the_stable_range:_classification#The Whitney invariant (for either n even or N orientable)|Whitney invariant]]) and because $\pi_2(SO_3)=0$. If $\varphi$ is a spin isomorphism, then $M_\varphi$ is spin \cite{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 $$y\cap C:=PD[(PDy)|_C]\in H_{k-2}(C,\partial C).$$ 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. \cite{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 $$A\cap C_f=A_f\quad\text{and}\quad A\cap C_{f'}=A_{f'}.$$ If $\varphi$ is a spin isomorphism and $W(f)=W(f')$, then there is a joint homology Seifert surface for $f$ and $f'$ \cite{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 $$\sigma_x(Q):=\frac{xPDp_1(Q)-x^3}3\in H_0(Q)=\Zz$$ 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 \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.) {{beginthm|Definition}} The ''Kreck invariant'' of two embeddings $f$ and $f'$ such that $W(f)=W(f')$ is defined by $$\eta(f,f'):\equiv\frac{\sigma_{2A}(M_\varphi)}{16}= \frac{APDp_1(M_\varphi)-4A^3}{24}\mod d(W(f)),$$ 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. \cite{Ekholm2001}, 4.1, \cite{Zhubr2009}. {{endthm}} We have A\mod2=0=PDw_2(M_\varphi)$, so any closed connected oriented 4-submanifold of $M_\varphi$ representing the class A$ 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. 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 \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 of the equations). See also [[T. Moriyama, Integral formula for an extension of Haeliger's embedding invariant]]. {{beginthm|The Kreck Invariant Lemma}}\label{th11}\cite{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 $$\frac{\sigma_{2[Y]}(M_\varphi)}{16}=\frac{\sigma(Y)-\overline p_1}8= \frac{\sigma(Y)-\overline e\cap\overline e}8.$$ {{endthm}} </wikitex> == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]] {{Stub}}n$-manifolds in $2n$-space, which are discussed in [Skopenkov2016e], [Skopenkov2006, $\S$2.4 `The Whitney invariant']. In this page we concentrate on more advanced classification results 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, $\S$1, $\S$3]. Unless specified otherwise, we work in the smooth category. For definition of the embedded connected sum $\#$ of embeddings of closed connected 3-manifolds $N$ in 6-space, and for the corresponding action of the group $E^6_D(S^3)$ on the set $E^6_D(N)$, see e.g. [Skopenkov2016c, $\S$4].

2 Examples

For any integer $a$ there is an embedding called the Hudson torus, $\Hud(a)\colon S^1\times S^2\to\Rr^6$, see [Skopenkov2016e, $\S$3], [Skopenkov2006, Example 2.10].

Piecewise smooth (PS) embedding and isotopy are defined in [Skopenkov2016f, Remark 1.1].

Example 2.1 (The Haefliger trefoil knot). There is a smooth embedding $t:S^3\to\Rr^6$ which is not smoothly isotopic to the standard embedding [Haefliger1962, Theorem 4.3], but is PS isotopic to the standard embedding (by the Zeeman Unknotting Spheres Theorem 2.3 of [Skopenkov2016c] and [Skopenkov2016f, Remark 1.1]).

This embedding represents a generator of $E^6_D(S^3)\cong\Zz$ [Haefliger1966, Theorem 5.16].

Let us construct the Haefliger (higher-dimensional) trefoil knot $t$ [Haefliger1962, $\S$4.1]. We start from the 3-dimensional Borromean rings (see Figure 6 of [Skopenkov2016h]), which are three disjoint 3-spheres in $\R^6$ defined as follows. For coordinates in $\Rr^6$ defined by $(x,y,z)=(x_1,x_2,y_1,y_2,z_1,z_2)$, the three 3-spheres are given by the following three systems of equations:

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

Figure 1: The construction of the trefoil from the Borromean rings

Take any orientations the 3-spheres [Haefliger1962, $\S$4.1] (the isotopy class of the Haefliger trefoil knot could potentially depend on these orientations, but this isotopy class would generate $E^6_D(S^3)\cong\Zz$ for any such choice). These orientations define an embedding $S^3 \sqcup S^3 \sqcup S^3 \to \R^6$ up to isotopy. The Haefliger trefoil $t$ is the embedded connected sum of the components of this embedding.

The construction of the Haefliger trefoil $t$ is analogous to the construction of the trefoil from the 1-dimensional Borromean rings, depicted in Figure 1. An intermediate stage of the construction, when two components are connected but the third remains disjoint, yields the Whitehead link $w$. The 1-dimensional picture can also be regarded as a schematic picture for the construction of the Haefliger trefoil from the 3-dimensional Borromean rings.

For higher-dimensional generalizations see [Skopenkov2016h, $\S$5] and [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

$$\displaystyle h:\Rr P^3\to S^5\subset\Cc^3\quad\text{by}\quad h[(x,y)]=(x^2,\sqrt2xy,y^2).$$

It is easy to check that $h$ is an embedding. (The image of this embedding in $\Cc^3$ is given by the equations $b^2=2ac$, $|a|^2+|b|^2+|c|^2=1$.)

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 the classical classification results just below the stable range, see [Skopenkov2016e, Theorem 2.1], [Skopenkov2006, Theorem 2.13].)

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

Recall that any 3-manifold embeds into $\Rr^6$ by the strong Whitney embedding theorem [Skopenkov2006, Theorem 2.2.a]. For the classical classification in the PL category see [Skopenkov2016e, Theorem 2.1], [Skopenkov2006, Theorem 2.13].

The following results of this subsection are proved in [Skopenkov2008] unless other references are given. Let $N$ be a closed connected oriented 3-manifold.

For the next theorem, the Whitney invariant $W$ and and the Kreck invariant $\eta_u$ are defined in [Skopenkov2016e, $\S$5] and in $\S$4 below. For an abelian group $G$ the divisibility of the zero element is zero, and the divisibility of $x\in G-\{0\}$ is $\max\{d\in\Zz\ | \ \text{there is }x_1\in G:\ x=dx_1\}$.

Theorem 3.2. (a) The Whitney invariant

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

is surjective.

(b) For any $a\in H_1(N)$ the Kreck invariant

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

is bijective, where $d(u)=0$ is the divisibility of the projection of $u$ to the free part of $H_1(N)$.

Although part (a) first appeared in [Skopenkov2008], it is (as opposed to (b)) a simple corollary of results by Hudson and Haefliger.

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, $\S$4], [Skopenkov2016e, Example 3.1].

See a higher-dimensional generalization [Skopenkov2016e, Theorem 6.3].

Part (a) was announced with a short outline of proof in [Hausmann1972], and proved in [Takase2006, Proof of Theorem 4.2 in p. 9 of the arxiv version]. For an alternative proof see [Skopenkov2008, Corollary (1) in p. 2 of the arxiv version].

Addendum 3.4. If $f:N\to\Rr^6$ and $g:S^3\to\Rr^6$ are embeddings, then

$$\displaystyle W(f\#g)=W(f)\quad\text{and}\quad\eta_{W(f)}(f\#g)\equiv\eta_{W(f)}(f)+\eta_0(g)\mod d(W(f)).$$

E. g. for $N=\Rr P^3$ the embedded connected sum action of $E^6_D(S^3)$ on $E^6_D(N)$ is free while for $N=S^1\times S^2$ we have part (a) of the following corollary.

Corollary 3.5. (a) There is an embedding $f:S^1\times S^2\to\Rr^6$ such that for any knot $g:S^3\to\Rr^6$ the embedding $f\# g$ is isotopic to $f$. (We can take as $f$ the Hudson torus $\Hud(1)$.)

(b) For any embedding $f:N\to\Rr^6$ such that $f(N)\subset\Rr^5$ (e.g. for the standard embedding

Tex syntax error

${\rm i}_{6,2}:S^1\times S^2\to\Rr^6$) and any 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 3.2 are as hard to prove as the general case of Theorem 3.2.)

For a related classification of some disconnected 3-manifolds in 6-space see [Skopenkov2016h, $\S$6].

4 The Kreck invariant

The Kreck invariant was invented by M. Kreck and appeared in [Skopenkov2008]. We work in the smooth category and use notation and conventions [Skopenkov2016c, $\S$3]. Let $N$ be a closed connected oriented 3-manifold and $f,f':N\to\Rr^6$ embeddings. Fix orientation on $\Rr^6$, and so on $\partial C_f,\partial C_{f'}$.

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 [Smale1959, Theorem A] 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 [Skopenkov2008, Spin Lemma]. Indeed, the restrictions to $N_0$ of $f$ and $f'$ are isotopic (this is proved in definition of the Whitney invariant [Skopenkov2016e, $\S$5], [Skopenkov2008, $\S$1, definition of the Whitney invariant]). Define $\varphi$ over $N_0$ using 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.

Identify with $\Zz$ the zero-dimensional homology group of a closed connected oriented manifold. The symbol of the intersection product $H_j(M)\times H_{6-j}(M)\to\Z$ in homology of 6-manifolds $M$ will be omitted.

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

$$\displaystyle Y\in H_4(M_\varphi)\quad\text{such that}\quad Y[\partial D^3_f]=1.$$

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

Remark 4.3. The homology Alexander Duality isomorphism $A_f:H_3(N)\to H_4(C_f,\partial)$ is defined in [Skopenkov2016f, $\S$4].

For $Y\in H_4(M_\varphi)$ denote $Y\cap C_f:=PD[(PDY)|_{C_f}]\in H_4(C_f,\partial)$. 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 $f'$ the classes

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

are `homology Seifert surfaces' for $f$, cf. \cite[ $\S$4 ]{Skopenkov2016f}. This property provides an equivalent definition of a joint Seifert class $Y$ which explains the name and which was used in [Skopenkov2008] together with the name `joint homology Seifert surface'.

Denote by $\sigma(X)$ the signature of a 4-manifold $X$. We use characteristic classes $w_2$ and $p_1$. 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 $H_4(Q)\cong H^2(Q)\cong[Q,\Cc P^\infty]$, 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 $\S$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.) Then the map $\eta_a:W^{-1}(a)\to \Zz_{d(a)}$ is well-defined by $\eta([f]):=\eta(f)$.

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, The Kreck Invariant Lemma in p. 7 of the arxiv version]. 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

• [Bolsinov&Fomenko2004] A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian systems, Chapman \& Hall/CRC, Boca Raton, FL, 2004. MR2036760 (2004j:37106) Zbl 1056.37075
• [Ekholm2001] T. Ekholm, Differential 3-knots in 5-space with and without self-intersections, Topology 40 (2001), no.1, 157–196. MR1791271 (2001h:57033) Zbl 0964.57029
• [Guillou&Marin1986] L. Guillou and A.Marin, Eds., A la r\'echerche de la topologie perdue, 1986, Progress in Math., 62, Birkhauser, Basel
• [Haefliger1962] A. Haefliger, Knotted $(4k-1)$-spheres in $6k$-space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
• [Haefliger1966] A. Haefliger, Differential embeddings of $S^n$ in $S^{n+q}$ for $q>2$, Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
• [Hausmann1972] J. Hausmann, Plongements de sphères d'homologie, C. R. Acad. Sci. Paris Sér. A-B 275 (1972), A963–965. MR0315727 (47 #4276) Zbl 0244.57005
• [Hirzebruch1966] F. Hirzebruch, Topological methods in algebraic geometry, Springer-Verlag, New York, 1966. MR0202713 (34 #2573) Zbl 0843.14009
• [Moriyama] T. Moriyama, Integral formula for an extension of Haefliger's embedding invariant, preprint.
• [Moriyama2008] T. Moriyama, An invariant of embeddings of 3-manifolds in 6-manifolds and Milnor's triple linking number, J. Math. Sci. Univ. Tokyo, 18 (2011), 193--237. arXiv:0806.3733.

• [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.
• [Skopenkov2008] A. Skopenkov, A classification of smooth embeddings of 3-manifolds in 6-space, Math. Z. 260 (2008), no.3, 647–672. Available at the arXiv:0603429MR2434474 (2010e:57028) Zbl 1167.57013

• [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
• [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
• [Skopenkov2016f] A. Skopenkov, 4-manifolds in 7-space, to appear in Bull. Man. Atl.
• [Skopenkov2016h] A. Skopenkov, High codimension links, to appear in Bull. Man. Atl.
• [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
• [Smale1959] S. Smale, Diffeomorphisms of the $2$-sphere, Proc. Amer. Math. Soc. 10 (1959), 621–626. MR0112149 (22 #3004) Zbl 0118.39103
• [Takase2004] M. Takase, A geometric formula for Haefliger knots, Topology 43 (2004), no.6, 1425–1447. MR2081431 (2005e:57032) Zbl 1060.57021
• [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

• [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).