4-manifolds in 7-space

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This page is being refereed under the supervision of the editorial board. Hence the page may not be edited at present. As always, the discussion page remains open for observations and comments.


1 Introduction

This page is intended not only for specialists in embeddings, but also for mathematician from other areas who want to apply or to learn the theory of embeddings.

Basic results on 4-manifolds in 7-space are particular cases of results on n-manifolds in (2n-1)-space for n=4 [Skopenkov2016e]. In this page we concentrate on more advanced results peculiar for n=4.

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, \S1, \S3].

2 Examples of knotted tori

There are the standard embeddings
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for p,q>0, p+q\le6 (which are defined in [Skopenkov2015a, \S2.1]).

The Hudson tori \Hud_{7,4,2}(a):S^2\times S^2\to S^7 and \Hud_{7,4,1}(a):S^1\times S^3\to S^7 are defined for an integer a in Remark 3.5.d of [Skopenkov2016e].

Denote by \eta:S^3\to S^2 the Hopf fibration and by pr_k the projection onto the k-th factor of a Cartesian product. Define D^m_+,D^m_-\subset S^m by the equations x_1\ge0 and x_1\le0, respectively.

Example 2.1 (Spinning construction). For an embedding g:S^3\to D^6 denote by Sg the embedding

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is the standard inclusion.

The restriction of Sg to D^1_+\times S^3 is isotopic to the standard embedding. We conjecture that if t:S^3\to D^6 is the Haefliger trefoil knot, then St is not smoothly isotopic to the connected sum of the standard embedding and any embedding S^4\to S^7.

The following Examples 2.2 and 2.3 appear in [Skopenkov2006] but could be known earlier.

Example 2.2. Two sembeddings \tau^1,\tau^2:S^1\times S^3\to S^7 are defined as compositions

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where k=1,2 and maps t^k:S^1\times S^3\to S^3 are defined below. We shall see that t^k|_{S^1\times y} is an embedding for each y\in S^3 and k=1,2, hence \tau^1 and \tau^2 are embeddings.

Define t^1(s,y):=sy, where S^3 is identified with the set of unit length quaternions and S^1\subset S^3 with the set of unit length complex numbers.

Define t^2(e^{i\theta},y):=\eta(y)\cos\theta+\sin\theta, where S^2 is identified with the 2-sphere formed by unit length quaternions of the form ai+bj+ck.

It would be interesting to know if \tau^2 is PL or smoothly isotopic to the Hudson torus \Hud_{7,4,1}(1).

Example 2.2 can be generalized as follows.

Example 2.3. Define a map \tau \colon \Z^2 \to E^7(S^1 \times S^3). Take a smooth map \alpha:S^3\to V_{4,2}. Assuming that V_{4, 2}\subset (\R^4)^2, we have \alpha(x) = (\alpha_1(x), \alpha_2(x)). Define the adjunction map \R^2 \times S^3 \to \R^4 by ((s, t), x) \mapsto \alpha_1(x)s + \alpha_2(x)t. (Assuming that V_{4, 2}\subset (\R^4)^{\R^2}, this map is obtained from \alpha by the exponential law.) Denote by \overline\alpha:S^1\times S^3\to S^3 the restriction of the adjunction map. We define the embedding \tau_\alpha to be the composition

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We define the map \tau by \tau(l, b):=[\tau_{\alpha}], where \alpha\colon S^3 \to V_{4, 2} represents (l, b) \in \pi_3(V_{4, 2}) (for the standard identification \pi_3(V_{4, 2})=\Zz^2).

Clearly, \tau^1=\tau(1,0) and \tau^2=\tau(0,1). See a generalization in [Skopenkov2016k].

It would be interesting to know if \tau(l,b) is smoothly or piecewise smoothly (PS) isotopic to \tau(l,b+2l) for each b,l\in\Zz.

We conjecture that

  • every PS embedding S^1\times S^3\to S^7 is PS isotopic to \tau(l,b) for some l,b\in\Z.
  • every smooth embedding S^1\times S^3\to S^7 is smoothly isotopic to \tau(l,b)\#g for some l,b\in\Z and embedding g:S^4\to S^7.

Example 2.4 (the Lambrechts torus). There is a smooth embedding S^1\times S^3\to S^7 whose complement is not homotopy equivalent to the complement of the standard embedding.

I learned this simple construction from P. Lambrechts. Take the Hopf fibration S^3\to S^7\overset{\nu}\to S^4. Take the Hopf linking h:S^1\sqcup S^2\to S^4 [Skopenkov2016h]. Then

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We conjecture that \nu^{-1}h(S^1)=im\tau^1.

Example 2.5 (the Haefliger torus [Boechat&Haefliger1970, p.165], [Boechat1971, 6.2]). There is a PL embedding S^2\times S^2\to S^7 which is (locally flat but) not PL isotopic to a smooth embedding.

Take the Haefliger trefoil knot S^3\to S^6. Extend it to a conical embedding D^4\to D^7_-. By [Haefliger1962], the trefoil knot also extends to a smooth embedding S^2\times S^2-Int D^4\to D^7_+ [Skopenkov2006, Figure 3.7.a]. These two extensions together form the Haefliger torus [Skopenkov2006, Figure 3.7.b].

3 Embeddings of the complex projective plabe

Example 3.1 [Boechat&Haefliger1970, p.164]. There is a smooth embedding \Cc P^2\to\Rr^7.

Recall that \Cc P^2_0 is the mapping cylinder of \eta. Recall that S^6=S^2*S^3. Define an embedding f:\Cc P^2_0\to S^6 by f[(x,t)]:=[(x,\eta(x),t)], where x\in S^3. In other words, the segment joining x\in S^3 and \eta(x)\in S^2 is mapped onto the arc in S^6 joining x to \eta(x). Clearly, the boundary 3-sphere of \Cc P^2_0 is standardly embedded into S^6. Hence f extends to an embedding \Cc P^2\to\Rr^7.

Apriori the above extension need not be unique (because it can be changed by embedded connected sum with an embedding g:S^4\to D^6). Surprisingly, it is unique, and in the smooth category is the only embedding \Cc P^2\to\Rr^7 (up to isotopy and a hyperplane reflection of \Rr^7).

Theorem 3.2. (a) There are exactly two smooth isotopy classes of smooth embeddings \Cc P^2\to\Rr^7 (differing by a hyperplane reflection of \Rr^7).

(b) For each pair of smooth embeddings f:\Cc P^2\to\Rr^7 and g:S^4\to\Rr^7 the embedding f\#g is smoothly isotopic to f.

(c) The Whitney invariant is a 1-1 correspondence E^7_{PL}(\Cc P^2)\to\Z.

Parts (a) and (b) are proved in [Skopenkov2005, Triviality Theorem (a)] or follow by Theorem 5.2 below. Part (c) follows by [Boechat&Haefliger1970], cf. a generalization presented in [Skopenkov2016e].

4 The Boechat-Haefliger invariant

Let N be a closed connected orientable 4-manifold and f:N\to\Rr^7 an embedding. Fix an orientation on N and an orientation on \Rr^7.

Definition 4.1. The composition

\displaystyle  H_{s+1}(C_f,\partial)\overset\partial\to H_s(\partial C_f)\overset{\nu_f}\to H_s(N)

of the boundary map \partial and the projection \nu_f is an isomorphism, cf. [Skopenkov2008, the Alexander Duality Lemma]. The inverse A_{f,s} to this composition is the homology Alexander Duality isomorphism; it equals to the composition H_s(N)\to H^{6-s}(C_f)\to H_{s+1}(C_f,\partial) of the cohomology Alexander and Poincaré duality isomorphisms.

Definition 4.2. A homology Seifert surface for f is the image A_{f,4}[N]\in H_5(C_f,\partial) of the fundamental class [N]. Define

\displaystyle  \varkappa(f):=A_{f,2}^{-1}\left(A_{f,4}[N]\cap A_{f,4}[N]\right)\in H_2(N).

Remark 4.3. (a) Take a small oriented disk D^3_f\subset\Rr^7 whose intersection with fN consists of exactly one point of sign +1 and such that \partial D^3_f\subset\partial C_f. A meridian of f is \partial D^3_f. A homology Seifert surface Y\in H_5(C_f,\partial) for f is uniquely defined by the condition Y\cap [\partial D^3_f]=1.

(b) We have \varkappa(f)-\varkappa(f_0)=\pm2W(f,f_0) for the Whitney invariant W(f,f_0) [Skopenkov2016e]. This is proved analogously to [Skopenkov2008, \S2, The Boechat-Haefliger Invariant Lemma].

(c) Definition 4.2 is equivalent to the original one [Boechat&Haefliger1970] by [Crowley&Skopenkov2008, Section Lemma 3.1]. Hence \varkappa(f)\mod2=PDw_2(N).

(d) Earlier notation was w_f [Boechat&Haefliger1970], BH(f) [Skopenkov2005] and \aleph(f) [Crowley&Skopenkov2008].

5 Classification

For the classification of E^7_{PL}(N) for a closed connected 4-manifold N with H_1(N)=0, see [Skopenkov2016e]. Here we work in the smooth category.

Theorem 5.1 ([Haefliger1966], see also [Skopenkov2005], [Crowley&Skopenkov2008]). There is an isomorphism E^7_D(S^4)\cong\Zz_{12}.

Theorem 5.2 ([Crowley&Skopenkov2008]). Let N be a closed connected 4-manifold such that H_1(N)=0. Then the image of the Boéchat-Haefliger invariant

\displaystyle \varkappa:E^7_D(N)\to H_2(N)
\displaystyle \text{is}\qquad im \varkappa=\{u\in H_2(N)\ |\ u\equiv PDw_2(N)\mod2,\ u\cap u=\sigma(N)\}.

For each u\in im \varkappa there is an injective invariant called the Kreck invariant,

\displaystyle \eta_u:\varkappa^{-1}(u)\to\Zz_{\gcd(u,24)}

whose image is the subset of even elements.

Here \gcd(u,24) is the maximal integer k such that both u\in H_2(N) and 24 are divisible by k. Thus \eta_u is surjective if u is not divisible by 2. Note that u\in im \varkappa is divisible by 2 (for some u or, equivalently, for each u) if and only if N is spin.

For the definition of the Kreck invariant see [Crowley&Skopenkov2008].

Corollary 5.3. (a) There are exactly twelve isotopy classes of embeddings N\to\Rr^7 if N is an integral homology 4-sphere (cf. Theorem 5.1).

(b) Identify H_2(S^2\times S^2) = \Z^2 using the standard basis. For each integer u there are exactly \gcd(u,12) isotopy classes of embeddings f:S^2\times S^2\to\Rr^7 with \varkappa(f)=(2u,0), and the same holds for those with \varkappa(f)=(0,2u). Other values of \Zz^2 are not in the image of \varkappa.

Addendum 5.4. Under the assumptions of Theorem 5.2 for each pair of embeddings f:N\to\Rr^7 and g:S^4\to\Rr^7

\displaystyle \varkappa(f\#g)=\varkappa(f)\quad\text{and}\quad\eta_{\varkappa(f)}(f\#g)\equiv\eta_{\varkappa(f)}(f)+\eta_0(g)\mod\gcd(\varkappa(f),24).

The following corollary gives examples where the embedded connected sum action of E^7_D(S^4) on E^7_D(N) is trivial and where it is effective.

Corollary 5.5. (a) Take an integer u and the Hudson torus f_u:=\Hud_{7,4,2}(u):S^2\times S^2\to\Rr^7 defined in Remark 3.5.d of [Skopenkov2016e]. If u=6k\pm1, then for each embedding g:S^4\to\Rr^7 the embedding f_u\#g is isotopic to f_u. Moreover, for a general integer u the number of isotopy classes of embeddings f_u\#g is \gcd(u,12).

(b) Let N be a closed connected 4-manifold such that H_1(N)=0 and the signature \sigma(N) of N is not divisible by the square of an integer s\ge2. Then for each pair of embeddings f:N\to\Rr^7 and g:S^4\to\Rr^7 the embedding f\#g is isotopic to f; in other words, \varkappa is injective.

(c) If N is a closed connected 4-manifold such that H_1(N)=0 and f(N)\subset\Rr^6 for an embedding f:N\to\Rr^7, then for every embedding g:S^4\to\Rr^7 the embedding f\#g is not isotopic to f.

We remark that Corollary 5.5(b) was first proved in [Skopenkov2005] independently of Theorem 5.2.

For classification when H_1(N)\ne0 see [Crowley&Skopenkov2016] and [Crowley&Skopenkov2016a].

6 References

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