4-manifolds in 7-space

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{{beginthm|Definition}}\label{dbh}
{{beginthm|Definition}}\label{dbh}
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Define `the Boechat-Haefliger invariant'
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Define `the Boechat-Haefliger invariant' of $f$
$$
$$
\varkappa(f):=A_{f,2}^{-1}\left(A_{f,4}[N]\cap A_{f,4}[N]\right)\in H_2(N).
\varkappa(f):=A_{f,2}^{-1}\left(A_{f,4}[N]\cap A_{f,4}[N]\right)\in H_2(N).
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$$
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$$
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Clearly, a map $\varkappa:E^7_D(N)\to H_2(N)$ is well-defined by $\varkappa([f]):=\varkappa(f)$.
{{endthm}}
{{endthm}}

Revision as of 12:43, 7 March 2019

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Contents

1 Introduction

Most of 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 embeddings of 4-manifolds in 7-space are particular cases of results on embeddings of n-manifolds in (2n-1)-space which is discussed in [Skopenkov2016e], [Skopenkov2006, \S2.4 `The Whitney invariant']. 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]. For definition of the embedded connected sum \# of embeddings of 4-manifolds N in 7-space, and for the corresponding action of the group E^7_D(S^4) on the set E^7_D(N), see e.g. [Skopenkov2016c, \S5]. Denote by \eta:S^3\to S^2 the Hopf fibration.

Remark 1.1 (PL and piecewise smooth embeddings). Any smooth manifold has a unique (up to PL homeomorphism) PL structure compatible with the given smooth structure [Milnor&Stasheff1974, Complement].

Since also any PL 4-manifold admits a unique smooth structure [Mandelbaum1980, \S1.2], we may consider a smooth 4-manifold as a PL 4-manifold.

A map of a smooth manifold is piecewise smooth (PS) if it is smooth on every simplex of some triangulation of the manifold. Clearly, every smooth or PL map is PS.

For a smooth manifold N let E^m_{PS}(N) be the set of PS embeddings N\to\R^m up to PS isotopy. The forgetful map E^m_{PL}(N)\to E^m_{PS}(N) is 1--1 [Haefliger1967, 2.4]. So a description of E^m_{PS}(N) is the same as a description of E^m_{PL}(N).

2 Examples of knotted tori

The natural standard embedding 
Tex syntax error
 for p,q>0, p+q\le6 is 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] or in [Skopenkov2006, Example 2.10].

Denote 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 

Tex syntax error
where 
Tex syntax error
 is the standard inclusion [Skopenkov2015a, \S2.1].

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 

Tex syntax error

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 

Tex syntax error

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)=\tau(l,b+2l) or [\tau(l,b)]=[\tau(l,b+2l)]\in E_{PS}^7(S^1\times S^3) for any b,l\in\Zz.

The unpublished papers [Crowley&Skopenkov2016], [Crowley&Skopenkov2016a] prove that

  • any PS embedding S^1\times S^3\to S^7 represents PS isotopy class of \tau(l,b) for some l,b\in\Z.
  • any smooth embedding S^1\times S^3\to S^7 represents smooth isotopy class of \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 

Tex syntax error
The last homotopy equivalence is proved in a more general form 
Tex syntax error
 for m\ge p+q+3 by the induction on p\ge0.

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 not PS 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 plane

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 
Tex syntax error
 joining x to \eta(x). Clearly, the boundary 3-sphere of \Cc P^2_0 is standardly embedded into 
Tex syntax error
. 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 composition with a hyperplane reflection of \Rr^7).

(b) For any 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 (defined in [Skopenkov2016e], [Skopenkov2006, \S2]) is a 1-1 correspondence E^7_{PL}(\Cc P^2)\to H_2(\Cc P^2)\cong \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.

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

Definition 4.2. Define `the Boechat-Haefliger invariant' of f

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

Clearly, a map \varkappa:E^7_D(N)\to H_2(N) is well-defined by \varkappa([f]):=\varkappa(f).

Remark 4.3. (a) Take a small oriented disk D^3_f\subset\Rr^7 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 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], [Skopenkov2006, \S2]. 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 is Poincaré dual to w_2(N) [Boechat&Haefliger1970].

(d) Earlier notation for \varkappa(f) 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], [Skopenkov2006, Theorem 2.13]. 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

  • PD:H^2(N)\to H_2(N) is the Poincaré duality isomorphism.
  • \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 any 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 any 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], [Skopenkov2006, Example 2.10]. If u=6k\pm1, then for any embedding g:S^4\to\Rr^7 the embedding f_u\#g is isotopic to f_u. Moreover, for any 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 any 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

, $\S]{Skopenkov2016c}. For definition of the [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Embedded_connected_sum|embedded connected sum]] $\#$ of embeddings of 4-manifolds $N$ in 7-space, and for the corresponding action of the group $E^7_D(S^4)$ on the set $E^7_D(N)$, see e.g. \cite[$\S]{Skopenkov2016c}. Denote by $\eta:S^3\to S^2$ the Hopf fibration. {{beginthm|Remark|(PL and piecewise smooth embeddings)}}\label{r:plps} Any smooth manifold has a unique (up to PL homeomorphism) PL structure compatible with the given smooth structure \cite[Complement]{Milnor&Stasheff1974}. Since also any PL 4-manifold admits a unique smooth structure \cite[$\S\S2.4 `The Whitney invariant']. 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]. For definition of the embedded connected sum \# of embeddings of 4-manifolds N in 7-space, and for the corresponding action of the group E^7_D(S^4) on the set E^7_D(N), see e.g. [Skopenkov2016c, \S5]. Denote by \eta:S^3\to S^2 the Hopf fibration.

Remark 1.1 (PL and piecewise smooth embeddings). Any smooth manifold has a unique (up to PL homeomorphism) PL structure compatible with the given smooth structure [Milnor&Stasheff1974, Complement].

Since also any PL 4-manifold admits a unique smooth structure [Mandelbaum1980, \S1.2], we may consider a smooth 4-manifold as a PL 4-manifold.

A map of a smooth manifold is piecewise smooth (PS) if it is smooth on every simplex of some triangulation of the manifold. Clearly, every smooth or PL map is PS.

For a smooth manifold N let E^m_{PS}(N) be the set of PS embeddings N\to\R^m up to PS isotopy. The forgetful map E^m_{PL}(N)\to E^m_{PS}(N) is 1--1 [Haefliger1967, 2.4]. So a description of E^m_{PS}(N) is the same as a description of E^m_{PL}(N).

2 Examples of knotted tori

The natural standard embedding 
Tex syntax error
 for p,q>0, p+q\le6 is 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] or in [Skopenkov2006, Example 2.10].

Denote 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 

Tex syntax error
where 
Tex syntax error
 is the standard inclusion [Skopenkov2015a, \S2.1].

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 

Tex syntax error

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 

Tex syntax error

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)=\tau(l,b+2l) or [\tau(l,b)]=[\tau(l,b+2l)]\in E_{PS}^7(S^1\times S^3) for any b,l\in\Zz.

The unpublished papers [Crowley&Skopenkov2016], [Crowley&Skopenkov2016a] prove that

  • any PS embedding S^1\times S^3\to S^7 represents PS isotopy class of \tau(l,b) for some l,b\in\Z.
  • any smooth embedding S^1\times S^3\to S^7 represents smooth isotopy class of \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 

Tex syntax error
The last homotopy equivalence is proved in a more general form 
Tex syntax error
 for m\ge p+q+3 by the induction on p\ge0.

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 not PS 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 plane

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 
Tex syntax error
 joining x to \eta(x). Clearly, the boundary 3-sphere of \Cc P^2_0 is standardly embedded into 
Tex syntax error
. 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 composition with a hyperplane reflection of \Rr^7).

(b) For any 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 (defined in [Skopenkov2016e], [Skopenkov2006, \S2]) is a 1-1 correspondence E^7_{PL}(\Cc P^2)\to H_2(\Cc P^2)\cong \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.

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

Definition 4.2. Define `the Boechat-Haefliger invariant' of f

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

Clearly, a map \varkappa:E^7_D(N)\to H_2(N) is well-defined by \varkappa([f]):=\varkappa(f).

Remark 4.3. (a) Take a small oriented disk D^3_f\subset\Rr^7 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 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], [Skopenkov2006, \S2]. 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 is Poincaré dual to w_2(N) [Boechat&Haefliger1970].

(d) Earlier notation for \varkappa(f) 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], [Skopenkov2006, Theorem 2.13]. 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

  • PD:H^2(N)\to H_2(N) is the Poincaré duality isomorphism.
  • \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 any 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 any 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], [Skopenkov2006, Example 2.10]. If u=6k\pm1, then for any embedding g:S^4\to\Rr^7 the embedding f_u\#g is isotopic to f_u. Moreover, for any 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 any 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

.2]{Mandelbaum1980}, we may consider a smooth 4-manifold as a PL 4-manifold. A map of a smooth manifold is ''piecewise smooth (PS)'' if it is smooth on every simplex of some triangulation of the manifold. Clearly, every smooth or PL map is PS. For a smooth manifold $N$ let $E^m_{PS}(N)$ be the set of PS embeddings $N\to\R^m$ up to PS isotopy. The forgetful map $E^m_{PL}(N)\to E^m_{PS}(N)$ is 1--1 \cite[2.4]{Haefliger1967}. So a description of $E^m_{PS}(N)$ is the same as a description of $E^m_{PL}(N)$. {{endthm}} == Examples of knotted tori== ; The natural ''standard embedding'' $\rm{{i}}:S^p\times S^q\to S^7$ for $p,q>0$, $p+q\le6$ is defined in \cite[$\S.1]{Skopenkov2015a}. 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 [[Embeddings_just_below_the_stable_range:_classification#Hudson_tori|Remark 3.5.d]] of \cite{Skopenkov2016e} or in \cite[Example 2.10]{Skopenkov2006}. Denote 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. {{beginthm|Example|(Spinning construction)}}\label{artin} For an embedding $g:S^3\to D^6$ denote by $Sg$ the embedding $$S^1\times S^3\overset{\id S^1\times g}\to S^1\times D^6\overset{\rm{i}}\to S^7,$$ where $\rm{i}$ is the standard inclusion \cite[$\S.1]{Skopenkov2015a}. {{endthm}} 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 [[3-manifolds_in_6-space#Examples|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 \ref{quat} and \ref{tau} appear in \cite{Skopenkov2006} but could be known earlier. {{beginthm|Example}}\label{quat} Two sembeddings $\tau^1,\tau^2:S^1\times S^3\to S^7$ are defined as compositions $$S^1\times S^3\overset{pr_2\times t^k}\to S^3\times S^3\overset{\rm{i}}\to S^7,$$ 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 $ai+bj+ck$. {{endthm}} It would be interesting to know if $\tau^2$ is PL or smoothly isotopic to the Hudson torus $\Hud_{7,4,1}(1)$. Example \ref{quat} can be generalized as follows. {{beginthm|Example}}\label{tau} 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 $$S^1\times S^3 \xrightarrow{\overline\alpha\times pr_2} S^3\times S^3 \xrightarrow{\rm{i}} S^7.$$ 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 $\pi_3(V_{4, 2})=\Zz^2$). {{endthm}} Clearly, $\tau^1=\tau(1,0)$ and $\tau^2=\tau(0,1)$. See [[Knotted_tori#Examples|a generalization]] in \cite{Skopenkov2016k}. It would be interesting to know if $\tau(l,b)=\tau(l,b+2l)$ or $[\tau(l,b)]=[\tau(l,b+2l)]\in E_{PS}^7(S^1\times S^3)$ for any $b,l\in\Zz$. The unpublished papers \cite{Crowley&Skopenkov2016}, \cite{Crowley&Skopenkov2016a} prove that * any PS embedding $S^1\times S^3\to S^7$ represents PS isotopy class of $\tau(l,b)$ for some $l,b\in\Z$. * any smooth embedding $S^1\times S^3\to S^7$ represents smooth isotopy class of $\tau(l,b)\#g$ for some $l,b\in\Z$ and embedding $g:S^4\to S^7$. {{beginthm|Example|(the Lambrechts torus)}}\label{lamb} 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 [[High_codimension_links#Examples|the Hopf linking]] $h:S^1\sqcup S^2\to S^4$ \cite{Skopenkov2016h}. Then $$\nu^{-1}h(S^1)\cong S^1\times S^3\quad\text{and}\quad S^7-\nu^{-1}h(S^1)\sim \nu^{-1}h(S^2)\cong S^2\times S^3\not\sim S^2\vee S^3\vee S^5\sim S^7-im \rm{i}.$$ The last homotopy equivalence is proved in a more general form $S^m-\rm{i}(S^p\times S^q)\sim S^{m-p-1}\vee S^{m-q-1}\vee S^{m-p-q-1}$ for $m\ge p+q+3$ by the induction on $p\ge0$. {{endthm}} We conjecture that $\nu^{-1}h(S^1)=im\tau^1$. {{beginthm|Example|(the Haefliger torus \cite[p.165]{Boechat&Haefliger1970}, \cite[6.2]{Boechat1971})}}\label{s2s2} There is a PL embedding $S^2\times S^2\to S^7$ which is not PS isotopic to a smooth embedding. Take [[3-manifolds_in_6-space#Examples|the Haefliger trefoil knot]] $S^3\to S^6$. Extend it to a conical embedding $D^4\to D^7_-$. By \cite{Haefliger1962}, the trefoil knot also extends to a smooth embedding $S^2\times S^2-Int D^4\to D^7_+$ \cite[Figure 3.7.a]{Skopenkov2006}. These two extensions together form the Haefliger torus \cite[Figure 3.7.b]{Skopenkov2006}. {{endthm}} == Embeddings of the complex projective plane== ; {{beginthm|Example|\cite[p.164]{Boechat&Haefliger1970}}}\label{ecp2} 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 $\Cc P^2\to\Rr^7$. {{endthm}} 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 $\Rr^7$). {{beginthm|Theorem}}\label{cp2} (a) There are exactly two smooth isotopy classes of smooth embeddings $\Cc P^2\to\Rr^7$ (differing by composition with a hyperplane reflection of $\Rr^7$). (b) For any 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) [[Embeddings_just_below_the_stable_range:_classification#The_Whitney_invariant|The Whitney invariant]] (defined in \cite{Skopenkov2016e}, \cite[$\S]{Skopenkov2006}) is a 1-1 correspondence $E^7_{PL}(\Cc P^2)\to H_2(\Cc P^2)\cong \Z$. {{endthm}} Parts (a) and (b) are proved in \cite[Triviality Theorem (a)]{Skopenkov2005} or follow by [[#Classification|Theorem \ref{clth4}]] below. Part (c) follows by \cite{Boechat&Haefliger1970}, cf. [[Embeddings_just_below_the_stable_range:_classification#A_generalization_to_highly-connected_manifolds|a generalization]] presented in \cite{Skopenkov2016e}. == 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 $\Rr^7$. {{beginthm|Definition}} The composition $$ H_{s+1}(C_f,\partial)\overset\partial\to H_s(\partial C_f)\overset{\nu_f}\to H_s(N) $$ of the boundary map $\partial$ and the projection $\nu_f$ is an isomorphism, cf. \cite[the Alexander Duality Lemma]{Skopenkov2008}. 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. A `homology Seifert surface' for $f$ is the image $A_{f,4}[N]\in H_5(C_f,\partial)$ of the fundamental class $[N]$. {{endthm}} {{beginthm|Definition}}\label{dbh} Define `the Boechat-Haefliger invariant' $$ \varkappa(f):=A_{f,2}^{-1}\left(A_{f,4}[N]\cap A_{f,4}[N]\right)\in H_2(N). $$ {{endthm}} {{beginthm|Remark}} (a) Take a small oriented disk $D^3_f\subset\Rr^7$ 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 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 [[Embeddings_just_below_the_stable_range#The Whitney invariant|the Whitney invariant]] $W(f,f_0)$ \cite{Skopenkov2016e}, \cite[$\S]{Skopenkov2006}. This is proved analogously to \cite[$\S, The Boechat-Haefliger Invariant Lemma]{Skopenkov2008}. (c) Definition \ref{dbh} is equivalent to the original one \cite{Boechat&Haefliger1970} by \cite[Section Lemma 3.1]{Crowley&Skopenkov2008}. Hence $\varkappa(f)\mod2$ is Poincaré dual to $w_2(N)$ \cite{Boechat&Haefliger1970}. (d) Earlier notation for $\varkappa(f)$ was $w_f$ \cite{Boechat&Haefliger1970}, $BH(f)$ \cite{Skopenkov2005} and $\aleph(f)$ \cite{Crowley&Skopenkov2008}. {{endthm}} == Classification == ; For the [[Embeddings_just_below_the_stable_range:_classification#Classification_just_below_the_stable_range|classification of $E^7_{PL}(N)$]] for a closed connected 4-manifold $N$ with $H_1(N)=0$, see \cite{Skopenkov2016e}, \cite[Theorem 2.13]{Skopenkov2006}. Here we work in the smooth category. {{beginthm|Theorem|(\cite{Haefliger1966}, see also \cite{Skopenkov2005}, \cite{Crowley&Skopenkov2008})}} \label{hae4} There is an isomorphism $E^7_D(S^4)\cong\Zz_{12}$. {{endthm}} {{beginthm|Theorem|(\cite{Crowley&Skopenkov2008})}}\label{clth4} Let $N$ be a closed connected 4-manifold such that $H_1(N)=0$. Then the image of the [[#The_Boechat-Haefliger_invariant|Boéchat-Haefliger invariant]] $$\varkappa:E^7_D(N)\to H_2(N)$$ $$\text{is}\qquad im \varkappa=\{u\in H_2(N)\ |\ u\equiv PDw_2(N)\mod2,\ u\cap u=\sigma(N)\}.$$ For each $u\in im \varkappa$ there is an injective invariant called the Kreck invariant, $$\eta_u:\varkappa^{-1}(u)\to\Zz_{\gcd(u,24)}$$ whose image is the subset of even elements. {{endthm}} Here * $PD:H^2(N)\to H_2(N)$ is the Poincaré duality isomorphism. * $\gcd(u,24)$ 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 \cite{Crowley&Skopenkov2008}. {{beginthm|Corollary}}\label{coclth4} (a) There are exactly twelve isotopy classes of embeddings $N\to\Rr^7$ if $N$ is an integral homology 4-sphere (cf. Theorem \ref{hae4}). (b) Identify $H_2(S^2\times S^2) = \Z^2$ using the standard basis. For any 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 $\varkappa$. {{endthm}} {{beginthm|Addendum}}\label{ad4} Under the assumptions of Theorem \ref{clth4} for any pair of embeddings $f:N\to\Rr^7$ and $g:S^4\to\Rr^7$ $$\varkappa(f\#g)=\varkappa(f)\quad\text{and}\quad\eta_{\varkappa(f)}(f\#g)\equiv\eta_{\varkappa(f)}(f)+\eta_0(g)\mod\gcd(\varkappa(f),24).$$ {{endthm}} 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. {{beginthm|Corollary}}\label{corclth4} (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 [[Embeddings_just_below_the_stable_range:_classification#Hudson_tori|Remark 3.5.d]] of \cite{Skopenkov2016e}, \cite[Example 2.10]{Skopenkov2006}. If $u=6k\pm1$, then for any embedding $g:S^4\to\Rr^7$ the embedding $f_u\#g$ is isotopic to $f_u$. Moreover, for any 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 any 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 $f$. {{endthm}} We remark that Corollary \ref{corclth4}(b) was first proved in \cite{Skopenkov2005} independently of Theorem \ref{clth4}. For classification when $H_1(N)\ne0$ see \cite{Crowley&Skopenkov2016} and \cite{Crowley&Skopenkov2016a}. == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]]\S2.4 `The Whitney invariant']. 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]. For definition of the embedded connected sum \# of embeddings of 4-manifolds N in 7-space, and for the corresponding action of the group E^7_D(S^4) on the set E^7_D(N), see e.g. [Skopenkov2016c, \S5]. Denote by \eta:S^3\to S^2 the Hopf fibration.

Remark 1.1 (PL and piecewise smooth embeddings). Any smooth manifold has a unique (up to PL homeomorphism) PL structure compatible with the given smooth structure [Milnor&Stasheff1974, Complement].

Since also any PL 4-manifold admits a unique smooth structure [Mandelbaum1980, \S1.2], we may consider a smooth 4-manifold as a PL 4-manifold.

A map of a smooth manifold is piecewise smooth (PS) if it is smooth on every simplex of some triangulation of the manifold. Clearly, every smooth or PL map is PS.

For a smooth manifold N let E^m_{PS}(N) be the set of PS embeddings N\to\R^m up to PS isotopy. The forgetful map E^m_{PL}(N)\to E^m_{PS}(N) is 1--1 [Haefliger1967, 2.4]. So a description of E^m_{PS}(N) is the same as a description of E^m_{PL}(N).

2 Examples of knotted tori

The natural standard embedding 
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 for p,q>0, p+q\le6 is 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] or in [Skopenkov2006, Example 2.10].

Denote 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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where 
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 is the standard inclusion [Skopenkov2015a, \S2.1].

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)=\tau(l,b+2l) or [\tau(l,b)]=[\tau(l,b+2l)]\in E_{PS}^7(S^1\times S^3) for any b,l\in\Zz.

The unpublished papers [Crowley&Skopenkov2016], [Crowley&Skopenkov2016a] prove that

  • any PS embedding S^1\times S^3\to S^7 represents PS isotopy class of \tau(l,b) for some l,b\in\Z.
  • any smooth embedding S^1\times S^3\to S^7 represents smooth isotopy class of \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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The last homotopy equivalence is proved in a more general form 
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 for m\ge p+q+3 by the induction on p\ge0.

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 not PS 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 plane

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 
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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 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 composition with a hyperplane reflection of \Rr^7).

(b) For any 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 (defined in [Skopenkov2016e], [Skopenkov2006, \S2]) is a 1-1 correspondence E^7_{PL}(\Cc P^2)\to H_2(\Cc P^2)\cong \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.

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

Definition 4.2. Define `the Boechat-Haefliger invariant' of f

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

Clearly, a map \varkappa:E^7_D(N)\to H_2(N) is well-defined by \varkappa([f]):=\varkappa(f).

Remark 4.3. (a) Take a small oriented disk D^3_f\subset\Rr^7 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 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], [Skopenkov2006, \S2]. 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 is Poincaré dual to w_2(N) [Boechat&Haefliger1970].

(d) Earlier notation for \varkappa(f) 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], [Skopenkov2006, Theorem 2.13]. 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

  • PD:H^2(N)\to H_2(N) is the Poincaré duality isomorphism.
  • \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 any 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 any 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], [Skopenkov2006, Example 2.10]. If u=6k\pm1, then for any embedding g:S^4\to\Rr^7 the embedding f_u\#g is isotopic to f_u. Moreover, for any 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 any 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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