3-manifolds in 6-space

Revision as of 09:32, 19 February 2010

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 $For notation and conventions 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.) $The$ $Kreck$ $invariant$ 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}. 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> == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]] {{Stub}} == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]] {{Stub}}t:S^3\to\Rr^6$ (which is a generator of $E^6(S^3)\cong\Zz$) [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].

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 Invariants

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

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

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 the definition of the Whitney invariant in classification just below the stable range) 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. 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 2.1. 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, [Zhubr].

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 [Moriyama, Integral formula for an extension of Haeliger's embedding invariant].

The Kreck Invariant Lemma 2.2.[Skopenkov2008] Let $f,f':N\to\Rr^6$ 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

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

3 Classification

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

Classification Theorem 3.1. The Whitney invariant

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

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 $\max\{d\in\Zz\ | \ \text{there is }x_1\in G:\ x=dx_1\}$. 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 and in some general remarks.

Addendum 3.3. 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

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

E. g. for $N=\Rr P^3$ 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.)

4 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
• [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

• [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

• [Zhubr] Template:Zhubr