4-manifolds in 7-space

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This page has been accepted for publication in the Bulletin of the Manifold Atlas.

Contents

1 Introduction

For notation and conventions see high codimension embeddings. Denote by \eta:S^3\to S^2 is the Hopf map.

2 Examples

There is The Hudson torus \Hud_{7,4,2}:S^2\times S^2\to\Rr^7.

Analogously to the case m=2n for an orientable 4-manifold N, an embedding f_0:N\to\Rr^7 and a class a\in H_1(N) one can construct an embedding f_a:N\to\Rr^7. However, this embedding is no longer well-defined. We have W(f_u,f_0)=u for the Whitney invariant (defined analogously to the case m=2n).

2.1 Embeddings of CP2 into R7

We follow [Boechat&Haefliger1970], p. 164. 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
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joining x to \eta(x). Clearly, the boundary 3-sphere of \Cc P^2_0 is standardly embedded into
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. Hence f extends to an embedding \Cc P^2\to\Rr^7.

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

Theorem 2.1.

  • There are exactly two isotopy classes of embeddings \Cc P^2\to\Rr^7 (differing by a hyperplane reflection of \Rr^7).
  • For each embeddings f:\Cc P^2\to\Rr^7 and g:S^4\to\Rr^7 the embedding f\#g is isotopic to f.

This follows by [Skopenkov2005], Triviality Theorem (a) or by general classification.

2.2 The Lambrechts torus and the Hudson torus

These two embeddings \tau^1,\tau^2:S^1\times S^3\to\Rr^7 are defined [Skopenkov2006] as compositions S^1\times S^3\overset{p_2\times t^i}\to S^3\times S^3\subset\Rr^7, where i=1,2, p_2 is the projection onto the second factor, \subset is the standard inclusion and maps t^i:S^1\times S^3\to S^3 are defined below. We shall see that t^i|_{S^1\times y} are embeddings for each y\in S^3, 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.

Note that \tau^2 is PL isotopic to the Hudson torus \Hud_{7,4,1}.

Take the Hopf fibration S^3\to S^7\overset\eta\to S^4. Take the standard embeding S^2\subset S^4. Its complement has the homotopy type of S^1. Then im\tau^1=\eta^{-1}(S^1)\cong S^1\times S^3\subset S^7. This is the construction of Lambrechts motivated by the following property:

\displaystyle S^7-im\tau^1\simeq \eta^{-1}(S^2)\cong S^2\times S^3\not\simeq S^2\vee S^3\vee S^5\simeq S^7-im f_0,

where f_0:S^1\times S^3\to S^7 is the standard embedding.

2.3 The Haefliger torus

This is a PL embedding S^2\times S^2\to\Rr^7 which is (locally flat but) not PL isotopic to a smooth embedding [Boechat&Haefliger1970], p.165, [Boechat1971], 6.2. Take the Haefliger trefoil knot S^3\to\Rr^6. Extend it to a conical embedding D^4\to\Rr^7_-. By [Haefliger1962], the trefoil knot also extends to a smooth embedding S^2\times S^2-Int D^4\to\Rr^7_+ (see [Skopenkov2006], Figure 3.7.a). These two extensions together form the Haefliger torus (see [Skopenkov2006], Figure 3.7.b).

3 The Boechat-Haefliger invariant

Let N be a closed connected orientable 4-manifold. Fix an orientation on N and an orientation on \Rr^7. A homology Seifert surface A_f for f is the image of the fundamental class [N] under the composition H_4(N)\to H^2(C_f)\to H_5(C_f,\partial C_f) of the Alexander and Poincar\'e duality isomorphisms. (This composition is an inverse to the composition H_5(C_f,\partial C_f)\to H_4(\partial C_f)\to H_4(N) of the boundary map \partial and the projection \nu_f of the normal bundle, cf. [Skopenkov2008], the Alexander Duality Lemma; this justifies the name `homology Seifert surface'.)

Define BH(f) to be the image of A_f^2=A_f\cap A_f under the composition H_3(C_f,\partial C_f)\to H^4(C_f)\to H_2(N) of the Poincar\'e and Alexander duality isomorphisms. (This composition has a direct geometric definition \nu\partial as above.)

This new definition is equivalent to the original one [Boechat&Haefliger1970] by [Crowley&Skopenkov2008], Section Lemma 3.1.

4 Classification

See a classification of E^7_{PL}(N) for a closed connected 4-manifold N such that H_1(N)=0. Here we work in the smooth category.

Theorem 4.1. E^7(S^4)\cong\Zz_{12}. [Haefliger1966], [Skopenkov2005], [Crowley&Skopenkov2008].

Theorem 4.2. [Crowley&Skopenkov2008] Let N be a closed connected 4-manifold such that H_1(N)=0. There is the Boéchat-Haefliger invariant

\displaystyle BH:E^7(N)\to H_2(N)

whose image is

\displaystyle im(BH)=\{u\in H_2(N)\ |\ u\equiv PDw_2(N)\mod2,\ u\cap u=\sigma(N)\}.

For each u\in im(BH) there is an injective invariant called the Kreck invariant,

\displaystyle \eta_u:BH^{-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(BH) 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].

Theorem 4.2 implies that

  • There are exactly twelve isotopy classes of embeddings N\to\Rr^7 if N is an integral homology 4-sphere (cf. Theorem 4.1).
  • For each integer u there are exactly GCD(u,12) isotopy classes of embeddings f:S^2\times S^2\to\Rr^7 with BH(f)=(2u,0), and the same holds for those with BH(f)=(0,2u). Other values of
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    are not in the image of BH. (We take the standard basis in H_2(S^2\times S^2).)

Theorem 4.2 implies the following examples (first proved in [Skopenkov2005]) of the triviality and the effectiveness of the connected sum action E^7(S^4)\to E^7(N).

  • 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 embeddings f:N\to\Rr^7 and g:S^4\to\Rr^7 the embedding f\#g is isotopic to f (in other words, BH is injective).
  • 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 each embedding g:S^4\to\Rr^7 the embedding f\#g is not isotopic to f.

The following can be obtained using [Crowley&Skopenkov2008] (but not using [Skopenkov2005]).

  • Take an integer u and the Hudson torus f_u:=\Hud_{7,4,2}(u):S^2\times S^2\to\Rr^7. If u=6k\pm1, then for each embedding g:S^4\to\Rr^7 the embedding f_u\#g is isotopic to f_u. (For a general integer u the number of isotopy classes of embeddings f_u\#g is GCD(u,12).)

Under assumptions of Theorem 4.2 for each pair of embeddings f:N\to\Rr^7 and g:S^4\to\Rr^7

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

5 References

This page has not been refereed. The information given here might be incomplete or provisional.

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