3-manifolds in 6-space

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Contents

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 n-manifolds in 2n-space, which are discussed in [Skopenkov2016e], [Skopenkov2006, \S2.4 `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, \S1, \S3]. 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, \S4].

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, \S3], [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, \S4.1]. 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, \S4.1] (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, \S5] 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].

Theorem 3.1 [Haefliger1966, Theorem 5.16]. There is an isomorphism E^6_D(S^3)\cong\Zz.

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, \S5] and in \S4 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, \S4], [Skopenkov2016e, Example 3.1].

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

Corollary 3.3. (a) If H_1(N)=0 (i.e. N is an integral homology sphere), then the Kreck invariant E^6_D(N)\to\Zz is a 1-1 correspondence.

(b) If H_2(N)=0 (i.e. N is a rational homology sphere, e.g. N=\Rr P^3), then E^6_D(N) is in (non-canonical) 1-1 correspondence with \Zz\times H_1(N). More precisely, the Whitney invariant W:E^6_D(N)\to H_1(N) is surjective, and every its preimage is in canonical 1-1 correspondence (given by the Kreck invariant) with \Zz.

(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 any 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 3.5 below.

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
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) 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, \S6].

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, \S3]. 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, \S5], [Skopenkov2008, \S1, 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, \S4].

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[ \S4 ]{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 \S9.2] 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

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).
, $\S]{Skopenkov2016c}. Unless specified otherwise, we work in the smooth category. For definition of the [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Embedded_connected_sum|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. \cite[$\S]{Skopenkov2016c}. == Examples == ; For any integer $a$ there is an embedding called the [[Embeddings_just_below_the_stable_range:_classification#Hudson_tori|Hudson torus]], $\Hud(a)\colon S^1\times S^2\to\Rr^6$, see \cite[$\S]{Skopenkov2016e}, \cite[Example 2.10]{Skopenkov2006}. [[4-manifolds_in_7-space#Introduction|Piecewise smooth (PS)]] embedding and isotopy are defined in \cite[Remark 1.1]{Skopenkov2016f}. {{beginthm|Example|(The Haefliger trefoil knot)}}\label{hatr} There is a smooth embedding $t:S^3\to\Rr^6$ which is not smoothly isotopic to the standard embedding \cite[Theorem 4.3]{Haefliger1962}, but is PS isotopic to the standard embedding (by the Zeeman [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting_theorems|Unknotting Spheres Theorem 2.3]] of \cite{Skopenkov2016c} and \cite[Remark 1.1]{Skopenkov2016f}). This embedding represents a generator of $E^6_D(S^3)\cong\Zz$ \cite[Theorem 5.16]{Haefliger1966}. Let us construct the Haefliger (higher-dimensional) trefoil knot $t$ \cite[$\S.1]{Haefliger1962}. We start from the 3-dimensional [[High_codimension_links#Examples beyond the metastable range|Borromean rings]] (see Figure 6 of \cite{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: $$\left\{\begin{array}{c} x=0\ |y|^2+2|z|^2=1\end{array}\right., \qquad \left\{\begin{array}{c} y=0\ |z|^2+2|x|^2=1\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} z=0\ |x|^2+2|y|^2=1. \end{array}\right.$$ [[Image:HT.jpg|thumb|400px|Figure 1: The construction of the trefoil from the Borromean rings]] Take any orientations the 3-spheres \cite[$\S.1]{Haefliger1962} (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 [[High_codimension_links#Examples beyond the metastable range|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 [[Knots,_i.e._embeddings_of_spheres#Examples|higher-dimensional generalizations]] see \cite[$\S]{Skopenkov2016h} and \cite{Skopenkov2016k}. {{endthm}} {{beginthm|Example|(The Hopf embedding of $\Rr P^3$ into $S^5$)}}\label{rp3} Represent $\Rr P^3=\{(x,y)\in\Cc^2\ |\ |x|^2+|y|^2=1\}/\pm1.$ Define $$h:\Rr P^3\to S^5\subset\Cc^3\quad\text{by}\quad h[(x,y)]=(x^2,\sqrt2xy,y^2).$$ It is easy to check that $h$ is an embedding. (The image of this embedding in $\Cc^3$ is given by the equations $b^2=2ac$, $|a|^2+|b|^2+|c|^2=1$.) {{endthm}} 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 [[Embeddings_just_below_the_stable_range:_classification#Classification|classification results just below the stable range]], see \cite{Skopenkov2016e|Theorem 2.1}, \cite[Theorem 2.13]{Skopenkov2006}.) {{beginthm|Example|(Algebraic embeddings from the theory of integrable systems)}}\label{hamil} 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[Chapter 14]{Bolsinov&Fomenko2004}. E.g. the following system of equations corresponds to the ''Euler integrability case'' \cite[Chapter 14]{Bolsinov&Fomenko2004}: $$R_1^2+R_2^2+R_3^2=c_1,\quad R_1S_1+R_2S_2+R_3S_3=c_2,\quad \frac{S_1^2}{A_1}+\frac{S_2^2}{A_2}+\frac{S_3^2}{A_3}=c_3,$$ where $R_i$ and $S_i$ are real variables while -manifolds in 2n-space, which are discussed in [Skopenkov2016e], [Skopenkov2006, \S2.4 `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, \S1, \S3]. 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, \S4].

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, \S3], [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, \S4.1]. 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, \S4.1] (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, \S5] 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].

Theorem 3.1 [Haefliger1966, Theorem 5.16]. There is an isomorphism E^6_D(S^3)\cong\Zz.

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, \S5] and in \S4 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, \S4], [Skopenkov2016e, Example 3.1].

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

Corollary 3.3. (a) If H_1(N)=0 (i.e. N is an integral homology sphere), then the Kreck invariant E^6_D(N)\to\Zz is a 1-1 correspondence.

(b) If H_2(N)=0 (i.e. N is a rational homology sphere, e.g. N=\Rr P^3), then E^6_D(N) is in (non-canonical) 1-1 correspondence with \Zz\times H_1(N). More precisely, the Whitney invariant W:E^6_D(N)\to H_1(N) is surjective, and every its preimage is in canonical 1-1 correspondence (given by the Kreck invariant) with \Zz.

(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 any 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 3.5 below.

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
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) 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, \S6].

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, \S3]. 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, \S5], [Skopenkov2008, \S1, 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, \S4].

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[ \S4 ]{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 \S9.2] 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

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).
== Classification == ; Recall that any 3-manifold embeds into $\Rr^6$ by the [[Wikipedia:Whitney_embedding_theorem|strong Whitney embedding theorem]] \cite[Theorem 2.2.a]{Skopenkov2006}. For the classical [[Embeddings just below the stable range: classification#Classification|classification in the PL category]] see \cite[Theorem 2.1]{Skopenkov2016e}, \cite[Theorem 2.13]{Skopenkov2006}. {{beginthm|Theorem|\cite[Theorem 5.16]{Haefliger1966}}} \label{hae3} There is an isomorphism $E^6_D(S^3)\cong\Zz$. {{endthm}} The following results of this subsection are proved in \cite{Skopenkov2008} unless other references are given. Let $N$ be a closed connected oriented 3-manifold. For the next theorem, [[Embeddings_just_below_the_stable_range:_classification#The Whitney invariant|the Whitney invariant]] $W$ and and [[#The Kreck invariant|the Kreck invariant]] $\eta_u$ are defined in \cite[$\S]{Skopenkov2016e} and in $\S$\ref{s:KI} below. For an abelian group $G$ the divisibility of the zero element is zero, and the divisibility of $x\in G-\{0\}$ is $\max\{d\in\Zz\ | \ \text{there is }x_1\in G:\ x=dx_1\}$. {{beginthm|Theorem}}\label{th7} (a) The Whitney invariant $$W:E^6_D(N)\to H_1(N)$$ is surjective. (b) For any $a\in H_1(N)$ the Kreck invariant $$\eta_u:W^{-1}(u)\to\Zz_{d(u)}$$ is bijective, where $d(u)=0$ is the divisibility of the projection of $u$ to the free part of $H_1(N)$. {{endthm}} Although part (a) first appeared in \cite{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 [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Embedded_connected_sum |unlinked]] and [[Embeddings_just_below_the_stable_range:_classification#Hudson_tori|linked]] embedded connected sum with embeddings $S^3\to\Rr^6$ \cite[$\S]{Skopenkov2016c}, \cite[Example 3.1]{Skopenkov2016e}. See a [[Embeddings_just_below_the_stable_range:_classification#Classification_2|higher-dimensional generalization]] \cite[Theorem 6.3]{Skopenkov2016e}. {{beginthm|Corollary}}\label{co8} (a) If $H_1(N)=0$ (i.e. $N$ is an integral homology sphere), then the Kreck invariant $E^6_D(N)\to\Zz$ is a 1-1 correspondence. (b) If $H_2(N)=0$ (i.e. $N$ is a rational homology sphere, e.g. $N=\Rr P^3$), then $E^6_D(N)$ is in (non-canonical) 1-1 correspondence with $\Zz\times H_1(N)$. More precisely, the Whitney invariant $W:E^6_D(N)\to H_1(N)$ is surjective, and every its preimage is in canonical 1-1 correspondence (given by the Kreck invariant) with $\Zz$. (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 any 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 \ref{co10} below. {{endthm}} Part (a) was announced with a short outline of proof in \cite{Hausmann1972}, and proved in \cite[Proof of Theorem 4.2 in p. 9 of the arxiv version]{Takase2006}. For an alternative proof see \cite[Corollary (1) in p. 2 of the arxiv version]{Skopenkov2008}. {{beginthm|Addendum}}\label{ad} If $f:N\to\Rr^6$ and $g:S^3\to\Rr^6$ are embeddings, then $$W(f\#g)=W(f)\quad\text{and}\quad\eta_{W(f)}(f\#g)\equiv\eta_{W(f)}(f)+\eta_0(g)\mod d(W(f)).$$ {{endthm}} 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. {{beginthm|Corollary}}\label{co10} (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 [[Embeddings_just_below_the_stable_range:_classification#Examples|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 ${\rm i}_{6,2}:S^1\times S^2\to\Rr^6$) and any non-trivial knot $g:S^3\to\Rr^6$ the embedding $f\# g$ is not isotopic to $f$. {{endthm}} (We believe that this very corollary or the case $N=\Rr P^3$ of Theorem \ref{th7} are as hard to prove as the general case of Theorem \ref{th7}.) For a related [[High_codimension_links#Linked_manifolds|classification of some disconnected 3-manifolds in 6-space]] see \cite[$\S]{Skopenkov2016h}. == The Kreck invariant == ; \label{s:KI} The Kreck invariant was invented by M. Kreck and appeared in \cite{Skopenkov2008}. We work in the smooth category and use [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Notation and conventions|notation and conventions]] \cite[$\S]{Skopenkov2016c}. 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 \cite[Theorem A]{Smale1959} this is equivalent to $\varphi$ being isotopic to the restriction of a vector bundle isomorphism to the spherical bundle.) {{beginthm|Definition}} For a bundle isomorphism $\varphi$ denote $$ M_\varphi:=C_f\cup_\varphi(-C_{f'}). $$ A bundle isomorphism $\varphi:\partial C_f\to\partial C_{f'}$ is called `spin', if $M_\varphi$ is spin. {{endthm}} A spin bundle isomorphism exists \cite[Spin Lemma]{Skopenkov2008}. Indeed, the restrictions to $N_0$ of $f$ and $f'$ are isotopic (this is proved in [[Embeddings_just_below_the_stable_range:_classification#The Whitney invariant|definition of the Whitney invariant]] \cite[$\S]{Skopenkov2016e}, \cite[$\Sn-manifolds in 2n-space, which are discussed in [Skopenkov2016e], [Skopenkov2006, \S2.4 `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, \S1, \S3]. 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, \S4].

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, \S3], [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, \S4.1]. 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, \S4.1] (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, \S5] 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].

Theorem 3.1 [Haefliger1966, Theorem 5.16]. There is an isomorphism E^6_D(S^3)\cong\Zz.

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, \S5] and in \S4 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, \S4], [Skopenkov2016e, Example 3.1].

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

Corollary 3.3. (a) If H_1(N)=0 (i.e. N is an integral homology sphere), then the Kreck invariant E^6_D(N)\to\Zz is a 1-1 correspondence.

(b) If H_2(N)=0 (i.e. N is a rational homology sphere, e.g. N=\Rr P^3), then E^6_D(N) is in (non-canonical) 1-1 correspondence with \Zz\times H_1(N). More precisely, the Whitney invariant W:E^6_D(N)\to H_1(N) is surjective, and every its preimage is in canonical 1-1 correspondence (given by the Kreck invariant) with \Zz.

(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 any 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 3.5 below.

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
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) 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, \S6].

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, \S3]. 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, \S5], [Skopenkov2008, \S1, 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, \S4].

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[ \S4 ]{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 \S9.2] 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

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).
, definition of the Whitney invariant]{Skopenkov2008}). 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_form|intersection product]] $H_j(M)\times H_{6-j}(M)\to\Z$ in homology of 6-manifolds $M$ will be omitted. {{beginthm|Definition}} 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 $$ Y\in H_4(M_\varphi)\quad\text{such that}\quad Y[\partial D^3_f]=1. $$ {{endthm}} If $W(f)=W(f')$ and $\varphi$ is a spin bundle isomorphism, then there is a joint Seifert class for $f,f'$ and $\varphi$ \cite[Agreement Lemma]{Skopenkov2008}. 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 $Q$). {{beginthm|Remark}} The [[4-manifolds_in_7-space#The_Boechat-Haefliger_invariant|homology Alexander Duality isomorphism]] $A_f:H_3(N)\to H_4(C_f,\partial)$ is defined in \cite[$\S]{Skopenkov2016f}. 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 $$ Y\cap C_f=A_f[N]\quad\text{and}\quad Y\cap C_{f'}=A_{f'}[N] $$ are `homology Seifert surfaces' for $f$, cf. \cite[ [[4-manifolds_in_7-space#The_Boechat-Haefliger_invariant|$\S]] ]{Skopenkov2016f}. This property provides an equivalent definition of a joint Seifert class $Y$ which explains the name and which was used in \cite{Skopenkov2008} together with the name `joint homology Seifert surface'. {{endthm}} Denote by $\sigma(X)$ the [[Intersection_form#Definition_of_signature|signature]] of a 4-manifold $X$. We use [[Stiefel-Whitney_characteristic_classes|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 $$\sigma_x(Q):=\frac{xPDp_1(Q)-x^3}3\in H_0(Q)=\Zz.$$ 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 \sigma(X)=PDp_1(X)=xPDp_1(Q)-x^3=3\sigma_x(Q)$ by \cite[end of $\S.2]{Hirzebruch1966} or else by \cite[Submanifold Lemma]{Skopenkov2008}. {{beginthm|Definition}} The `Kreck invariant' of two embeddings $f$ and $f'$ such that $W(f)=W(f')$ is defined by $$\eta(f,f'):=\rho_d\frac{\sigma_{2Y}(M_\varphi)}{16}=\rho_d\frac{PDp_1(M_\varphi)Y-4Y^3}{24}\in\Zz_d,$$ 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. \cite[4.1]{Ekholm2001}, \cite{Zhubr2009}. {{endthm}} We have Y\mod2=0=PDw_2(M_\varphi)$, so any closed connected oriented 4-submanifold of $M_\varphi$ representing the class Y$ is spin, hence by the Rokhlin Theorem $\sigma_{2Y}(M_\varphi)$ is indeed divisible by 16. The Kreck invariant is well-defined by \cite[Independence Lemma]{Skopenkov2008}. 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 \cite[Remarks to the four articles of Rokhlin, II.2.7 and III.excercises.IV.3]{Guillou&Marin1986}, \cite[Corollary 6.5]{Takase2004}, \cite[Proposition 4.1]{Takase2006}. 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 \cite{Moriyama}, \cite{Moriyama2008}. {{beginthm|The Kreck Invariant Lemma|\cite[The Kreck Invariant Lemma in p. 7 of the arxiv version]{Skopenkov2008}}}\label{th11} 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 $$\frac{\sigma_{2[Y]}(M_\varphi)}{16}=\frac{\sigma(Y)-\overline p_1}8= \frac{\sigma(Y)-\overline e\cap\overline e}8.$$ {{endthm}}
== References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]]n-manifolds in 2n-space, which are discussed in [Skopenkov2016e], [Skopenkov2006, \S2.4 `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, \S1, \S3]. 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, \S4].

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, \S3], [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, \S4.1]. 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, \S4.1] (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, \S5] 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].

Theorem 3.1 [Haefliger1966, Theorem 5.16]. There is an isomorphism E^6_D(S^3)\cong\Zz.

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, \S5] and in \S4 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, \S4], [Skopenkov2016e, Example 3.1].

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

Corollary 3.3. (a) If H_1(N)=0 (i.e. N is an integral homology sphere), then the Kreck invariant E^6_D(N)\to\Zz is a 1-1 correspondence.

(b) If H_2(N)=0 (i.e. N is a rational homology sphere, e.g. N=\Rr P^3), then E^6_D(N) is in (non-canonical) 1-1 correspondence with \Zz\times H_1(N). More precisely, the Whitney invariant W:E^6_D(N)\to H_1(N) is surjective, and every its preimage is in canonical 1-1 correspondence (given by the Kreck invariant) with \Zz.

(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 any 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 3.5 below.

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
) 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, \S6].

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, \S3]. 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, \S5], [Skopenkov2008, \S1, 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, \S4].

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[ \S4 ]{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 \S9.2] 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

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).
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