3-manifolds in 6-space

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For notation and conventions throughout this page see high codimension embeddings.

1 Examples

1.1 The Haefliger trefoil knot

Let us construct a smooth embedding $t:S^3\to\Rr^6$$For notation and conventions throughout this page see [[High_codimension_embeddings|high codimension embeddings]]. == Examples == === The Haefliger trefoil knot === <wikitex>; Let us construct a smooth embedding $t:S^3\to\Rr^6$ (which is a generator of $E^6(S^3)\cong\Zz$) \cite{Haefliger1962}, 4.1. A miraculous property of this embedding is that it is not $smoothly$ isotopic to the standard embedding, but is $piecewise$ $smoothly$ isotopic to the standard embedding. Denote coordinates in $\Rr^6$ by $(x,y,z)=(x_1,x_2,y_1,y_2,z_1,z_2)$. The higher-dimensional trefoil knot $t$ is obtained by joining with two tubes the higher-dimensional $Borromean$ $rings$, i.e. the three spheres given by the following three systems of equations: $$\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..$$ See Figures 3.5 and 3.6 of \cite{Skopenkov2006}. </wikitex> === The Hopf construction of an embedding <wikitex> $\Rr P^3\to S^5$ </wikitex> === <wikitex>; Represent $\Rr P^3=\{(x,y)\in\Cc^2\ |\ |x|^2+|y|^2=1\}/\pm1.$ Define $$Ho:\Rr P^3\to S^5\subset\Cc^3\quad\text{by}\quad Ho[(x,y)]=(x^2,2xy,y^2).$$ It is easy to check that $Ho$ is an embedding. (The image of this embedding is given by the equations $b^2=4ac$, $|a|^2+|b|^2+|c|^2=1$.) It would be interesting to obtain an explicit construction of an embedding $f:\Rr P^3\to S^6$ which is not isotopic to the composition of the Hopf embedding with the standard inclusion $S^5\subset\Rr^6$. (Such an embedding $f$ is unique up to PL isotopy by [[Classification_just_below_the_stable_range|classification just below the stable range]].) </wikitex> === Algebraic embeddings from the theory of integrable systems === <wikitex>; Some 3-manifolds appear in the theory of integrable systems together with their embeddings into $\Rr^6$ (given by a system of algebraic equations) \cite{Bolsinov&Fomenko2004}, Chapter 14. E.g. the following system of equations corresponds to the Euler integrability case: $$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,$$ where $R_i$ and $S_i$ are variables while $A_1<A_2<A_3$ and $c_i$ are constants. This defines embeddings of $S^3$, $S^1\times S^2$ or $\Rr P^3$ into $\Rr^6$. </wikitex> == Invariants == <wikitex>; <!--For notation and conventions see [[High_codimension_embeddings|high codimension embeddings]]. For classification theorem involving the Kreck invariant see [[Classification_of_embeddings_of_3-manifolds_in_the_6-space|classification of embeddings of 3-manifolds in the 6-space]]. --> Let $N$ be a closed connected orientable 3-manifold. An orientation-preserving diffeomorphism $\varphi:\partial C_f\to\partial C_{f'}$ such that $\nu_f=\nu_{f'}\varphi$ is simply called an $isomorphism$. For an isomorphism $\varphi$ denote $$M=M_\varphi:=C_f\cup_\varphi(-C_{f'}).$$ An isomorphism $\varphi:\partial C_f\to\partial C_{f'}$ is called $spin$, if $\varphi$ over $N_0$ is defined by an isotopy between the restrictions of $f$ and $f'$ to $N_0$. A spin isomorphism exists because the restrictions to $N_0$ of $f$ and $f'$ are isotopic (see the definition of the Whitney invariant in [[Classification_just_below_the_stable_range|classification just below the stable range]]) and because $\pi_2(SO_3)=0$. If $\varphi$ is a spin isomorphism, then $M_\varphi$ is spin \cite{Skopenkov2008}, Spin Lemma. Denote by $\sigma (X)$ the signature of a 4-manifold $X$. Denote by $PD:H^i(Q)\to H_{q-i}(Q,\partial Q)$ and $PD:H_i(Q)\to H^{q-i}(Q,\partial Q)$ Poincar\'e duality (in any manifold $Q$). For $y\in H_4(M_\varphi)$ and a $k$-submanifold $C\subset M_\varphi$ (e.g. $C=C_f$ or $C=\partial C_f$) denote $$y\cap C:=PD[(PDy)|_C]\in H_{k-2}(C,\partial C).$$ If $y$ is represented by a closed oriented 4-submanifold $Y\subset M_\varphi$ in general position to $C$, then $y\cap C$ is represented by $Y\cap C$. A $homology$ $Seifert$ $surface$ for $f$ is the image $A_f$ of the fundamental class $[N]$ under the composition $H_3(N)\to H^2(C_f)\to H_4(C_f,\partial C_f)$ of the Alexander and Poincar\'e duality isomorphisms. A $joint$ $homology$ $Seifert$ $surface$ for $f$ and $f'$ is a class $A\in H_4(M_\varphi)$ such that $$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{Zhubr}. {{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 [[http://www.ms.u-tokyo.ac.jp/~tetsuhir/mathematics/Integral-Formula.pdf|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.$$ {{endthm}} </wikitex> == Classification == <wikitex>; <!--For notation and conventions see [[High_codimension_embeddings|high codimension embeddings]].--> The results of this page are proved in \cite{Skopenkov2008} unless other references are given. Let $N$ be a closed connected orientable 3-manifold. {{beginthm|Classification Theorem}}\label{th7} The Whitney invariant $$W:E^6_D(N)\to H_1(N)$$ is surjective. For each $a\in H_1(N)$ the Kreck invariant $$\eta_a:W^{-1}(u)\to\Zz_{d(a)}$$ is bijective, where $d(a)$ is the divisibility of the projection of $a$ to the free part of $H_1(N)$. {{endthm}} Recall that for an abelian group $G$ the divisibility of zero is zero and the divisibility of $x\in G-\{0\}$ is $\max\{d\in\Zz\ | \ \text{there is }x_1\in G:\ x=dx_1\}$. [[Definition_of_the_Kreck_invariant_for_3-manifolds_in_the_6-space|Definition of the Kreck invariant]]. All isotopy classes of embeddings $N\to\Rr^6$ can be constructed (from a certain given embedding) using connected sum with embeddings $S^3\to\Rr^6$, see the construction in [[Classification_just_below_the _stable_range|classification just below the stable range]] and in [[Some_general_remarks|some general remarks]]. {{beginthm|Corollary}}\label{co8} *The Kreck invariant $\eta_0:E^6(N)\to\Zz$ is a 1--1 correspondence if $N$ is $S^3$ or an integral homology sphere \cite{Haefliger1966}, \cite{Hausmann1972}, \cite{Takase2006}. (For $N=S^3$ the Kreck invariant is also a group isomorphism; this does not follows from the Classification Theorem \ref{th7} but is proved in \cite{Haefliger1966}.) *If $H_2(N)=0$ (i.e. $N$ is a rational homology sphere, e.g. $N=\Rr P^3$), then $E^6(N)$ is in (non-canonical) 1--1 correspondence with $\Zz\times H_1(N)$. *Embeddings $S^1\times S^2\to\Rr^6$ with zero Whitney invariant are in 1--1 correspondence with $\Zz$, and for each integer $k\ne0$ there are exactly $k$ distinct (i.e. non-isotopic) embeddings $S^1\times S^2\to\Rr^6$ with the Whitney invariant $k$, cf. Corollary \ref{co10} below. *The Whitney invariant $W:E^6(N_1\# N_2)\to H_1(N_1\# N_2)\cong H_1(N_1)\oplus H_1(N_2)$ is surjective and $\# W^{-1}(a_1\oplus a_2)=\#\Zz_{GCD(d(a_1),d(a_2))}$. {{endthm}} {{beginthm|Addendum}}\label{ad9} If $f:N\to\Rr^6$ is an embedding, $t$ is the generator of $E^6(S^3)\cong\Zz$ and $kt$ is a connected sum of $k$ copies of $t$, then $$W(f\#kt)=W(f)\quad and\quad \eta_{W(f)}(f\#kt)\equiv\eta_{W(f)}(f)+k\mod d(W(f)).$$ {{endthm}} E. g. for $N=\Rr P^3$ the `connected sum' action of $E^6(S^3)$ on $E^6(N)$ is free while for $N=S^1\times S^2$ we have the following corollary. (We believe that this very concrete corollary or the case $N=\Rr P^3$ are as non-trivial as the general case of the Classification Theorem.) {{beginthm|Corollary}}\label{co10} *There is an embedding $f=\Hud(3):S^1\times S^2\to\Rr^6$ such that for each knot $g:S^3\to\Rr^6$ the embedding $f\# g$ is isotopic to $f$. *For each embedding $f:N\to\Rr^6$ such that $f(N)\subset\Rr^5$ (e.g. for the standard embedding $f:S^1\times S^2\to\Rr^6$) and each non-trivial knot $g:S^3\to\Rr^6$ the embedding $f\# g$ is not isotopic to $f$. {{endthm}} </wikitex> == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]] {{Stub}}t:S^3\to\Rr^6$ (which is a generator of $E^6(S^3)\cong\Zz$$E^6(S^3)\cong\Zz$) [Haefliger1962], 4.1. A miraculous property of this embedding is that it is not $smoothly$$smoothly$ isotopic to the standard embedding, but is $piecewise$$piecewise$ $smoothly$$smoothly$ isotopic to the standard embedding.

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 higher-dimensional trefoil knot $t$$t$ is obtained by joining with two tubes the higher-dimensional $Borromean$$Borromean$ $rings$$rings$, i.e. the three spheres given by the following three systems of equations:

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

See Figures 3.5 and 3.6 of [Skopenkov2006].

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

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

It is easy to check that $Ho$$Ho$ is an embedding. (The image of this embedding is given by the equations $b^2=4ac$$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 S^6$$f:\Rr P^3\to S^6$ which is not isotopic to the composition of the Hopf embedding with the standard inclusion $S^5\subset\Rr^6$$S^5\subset\Rr^6$. (Such an embedding $f$$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$$\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:

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

where $R_i$$R_i$ and $S_i$$S_i$ are variables while $A_1<A_2<A_3$$A_1<A_2<A_3$ and $c_i$$c_i$ are constants. This defines embeddings of $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$.

2 Invariants

Let $N$$N$ be a closed connected orientable 3-manifold. An orientation-preserving diffeomorphism $\varphi:\partial C_f\to\partial C_{f'}$$\varphi:\partial C_f\to\partial C_{f'}$ such that $\nu_f=\nu_{f'}\varphi$$\nu_f=\nu_{f'}\varphi$ is simply called an $isomorphism$$isomorphism$. For an isomorphism $\varphi$$\varphi$ denote

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

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

Denote by $\sigma (X)$$\sigma (X)$ the signature of a 4-manifold $X$$X$. 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)$ Poincar\'e duality (in any manifold $Q$$Q$). For $y\in H_4(M_\varphi)$$y\in H_4(M_\varphi)$ and a $k$$k$-submanifold $C\subset M_\varphi$$C\subset M_\varphi$ (e.g. $C=C_f$$C=C_f$ or $C=\partial C_f$$C=\partial C_f$) denote

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

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

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

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

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

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

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

the virtual signature of $(Q,x)$$(Q,x)$. (Since $H_4(Q)\cong[Q,\Cc P^\infty]$$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)=p_1(X)=xPDp_1(Q)-x^3=3\sigma_x(Q)$$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 2.1. The Kreck invariant of two embeddings $f$$f$ and $f'$$f'$ such that $W(f)=W(f')$$W(f)=W(f')$ is defined by

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

where $\varphi:\partial C_f\to\partial C_{f'}$$\varphi:\partial C_f\to\partial C_{f'}$ is a spin isomorphism and $A\in H_4(M)$$A\in H_4(M)$ is a joint homology Seifert surface for $f$$f$ and $f'$$f'$. Cf. [Ekholm2001], 4.1, [Zhubr].

We have $2A\mod2=0=PDw_2(M_\varphi)$$2A\mod2=0=PDw_2(M_\varphi)$, so any closed connected oriented 4-submanifold of $M_\varphi$$M_\varphi$ representing the class $2A$$2A$ is spin, hence by the Rokhlin Theorem $\sigma_{2A}(M_\varphi)$$\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)$$a\in H_1(N)$ fix an embedding $f':N\to\Rr^6$$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, Integral formula for an extension of Haeliger's embedding invariant].

The Kreck Invariant Lemma 2.2.[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'}$$\varphi:\partial C_f\to\partial C_{f'}$ a spin isomorphism, $Y\subset M_\varphi$$Y\subset M_\varphi$ a closed connected oriented 4-submanifold representing a joint homology Seifert surface and $\overline p_1\in\Zz$$\overline p_1\in\Zz$, $\overline e\in H_2(Y)$$\overline e\in H_2(Y)$ are the Pontryagin number and Poincar\'e dual of the Euler classes of the normal bundle of $Y$$Y$ in $M_\varphi$$M_\varphi$. Then

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

3 Classification

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

Classification Theorem 3.1. The Whitney invariant

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

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

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

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)$.

Recall that for an abelian group $G$$G$ the divisibility of zero is zero and the divisibility of $x\in G-\{0\}$$x\in G-\{0\}$ is $\max\{d\in\Zz\ | \ \text{there is }x_1\in G:\ x=dx_1\}$$\max\{d\in\Zz\ | \ \text{there is }x_1\in G:\ x=dx_1\}$. Definition of the Kreck invariant. All isotopy classes of embeddings $N\to\Rr^6$$N\to\Rr^6$ can be constructed (from a certain given embedding) using connected sum with embeddings $S^3\to\Rr^6$$S^3\to\Rr^6$, see the construction in classification just below the stable range and in some general remarks.

Corollary 3.2.

• 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$$S^3$ or an integral homology sphere [Haefliger1966], [Hausmann1972], [Takase2006]. (For $N=S^3$$N=S^3$ the Kreck invariant is also a group isomorphism; this does not follows from the Classification Theorem 3.1 but is proved in [Haefliger1966].)
• 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)$$\Zz\times H_1(N)$.
• Embeddings $S^1\times S^2\to\Rr^6$$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$$k$ distinct (i.e. non-isotopic) embeddings $S^1\times S^2\to\Rr^6$$S^1\times S^2\to\Rr^6$ with the Whitney invariant $k$$k$, cf. Corollary 3.4 below.
• The Whitney invariant $W:E^6(N_1\# N_2)\to H_1(N_1\# N_2)\cong H_1(N_1)\oplus H_1(N_2)$$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 $\# W^{-1}(a_1\oplus a_2)=\#\Zz_{GCD(d(a_1),d(a_2))}$$\# W^{-1}(a_1\oplus a_2)=\#\Zz_{GCD(d(a_1),d(a_2))}$.

Addendum 3.3. If $f:N\to\Rr^6$$f:N\to\Rr^6$ is an embedding, $t$$t$ is 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

$\displaystyle W(f\#kt)=W(f)\quad and\quad \eta_{W(f)}(f\#kt)\equiv\eta_{W(f)}(f)+k\mod d(W(f)).$ $$\displaystyle W(f\#kt)=W(f)\quad and\quad \eta_{W(f)}(f\#kt)\equiv\eta_{W(f)}(f)+k\mod d(W(f)).$$

E. g. for $N=\Rr P^3$$N=\Rr P^3$ the `connected sum' action of $E^6(S^3)$$E^6(S^3)$ on $E^6(N)$$E^6(N)$ is free while for $N=S^1\times S^2$$N=S^1\times S^2$ we have the following corollary. (We believe that this very concrete corollary or the case $N=\Rr P^3$$N=\Rr P^3$ are as non-trivial as the general case of the Classification Theorem.)

Corollary 3.4.

• There is an embedding $f=\Hud(3):S^1\times S^2\to\Rr^6$$f=\Hud(3):S^1\times S^2\to\Rr^6$ such that for each knot $g:S^3\to\Rr^6$$g:S^3\to\Rr^6$ the embedding $f\# g$$f\# g$ is isotopic to $f$$f$.
• 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$.

4 References

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

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