4-manifolds in 7-space

(Difference between revisions)

Jump to: navigation, search

Revision as of 11:26, 28 March 2019

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

The user responsible for this page is Askopenkov. No other user may edit this page at present.

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 ${{Authors|Askopenkov}} == Introduction == <wikitex>; 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_just_below_the_stable_range:_classification#Classification_just_below_the_stable_range| embeddings of $n$-manifolds in $(2n-1)$-space]] which is discussed in \cite{Skopenkov2016e}, \cite[$\S.4 `The Whitney invariant']{Skopenkov2006}. In this page we concentrate on more advanced results peculiar for $n=4$. For a [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|general introduction to embeddings]] as well as the [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Notation and conventions|notation and conventions]] used on this page, we refer to \cite[$\Sn$ -manifolds in $(2n-1)$ -space which is discussed in [Skopenkov2016e], [Skopenkov2006, $\S$ 2.4 `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, $\S$ 1, $\S$ 3]. 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, $\S$ 5].

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, $\S$ 1.2], 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 normal framing on the standard embedding

Tex syntax error

${\rm i}:S^q\to S^m$ defines the standard embedding

Tex syntax error

${\rm i}_{m,q}:D^{m-q}\times S^q\to S^m$ . Denote by the same symbol

Tex syntax error

${\rm i}_{m,q}$ the restrictions of

Tex syntax error

${\rm i}_{m,q}$ to $S^p\times S^q$ $S^p\times S^q$ for $p\le m-q-1$ $p\le m-q-1$ .

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.

For an embedding $g:S^3\to D^6$ denote by $Sg$ the embedding

Tex syntax error

$\displaystyle S^1\times S^3\overset{g\times\id S^1}\to D^6\times S^1\overset{{\rm i}_{7,1}}\to S^7.$

The restriction of $Sg$ to $D^1_+\times S^3$ is isotopic to (the restriction to $D^1_+\times S^3$ of) 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.

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

Tex syntax error

$\displaystyle S^1\times S^3\overset{pr_2\times t^k}\to S^3\times S^3\overset{{\rm i}_{7,3}}\to S^7,$

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 $\eta:S^3\to S^2$ is the Hopf fibration and $S^2$ 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.

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

$\displaystyle S^1\times S^3 \xrightarrow{\overline\alpha\times pr_2} S^3\times S^3 \xrightarrow{{\rm i}_{7,3}} S^7.$

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 $\pi_3(V_{4, 2})=\Zz^2$ ).

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 $\tau(l,b)\#g$ for some $l,b\in\Z$ and $g\in E^7(S^4)$ .

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

$\displaystyle \nu^{-1}h(S^1)\cong S^1\times S^3\quad\text{and}\quad S^7-\nu^{-1}h(S^1)\sim \nu^{-1}h(S^2)\cong S^2\times S^3\not\sim S^2\vee S^3\vee S^5\sim S^7-{\rm i}_{7,3}(S^1\times S^3).$

The last homotopy equivalence is proved in a more general form

Tex syntax error

$S^m-{\rm i}_{m,q}(S^p\times S^q)\sim S^{m-p-1}\vee S^{m-q-1}\vee S^{m-p-q-1}$ for $m\ge p+q+3$ $m\ge p+q+3$ by induction on $p\ge0$ $p\ge0$ using the following observation: if $f:N\to S^n$ $f:N\to S^n$ is an embedding, then

Tex syntax error

$C_{{\rm i}\circ f}\sim\Sigma C_f$ .

We conjecture that $\nu^{-1}h(S^1)=\tau^1(S^1\times S^3)$ .

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

There is a smooth embedding $\Cc P^2\to\Rr^7$ .

Recall that $\Cc P^2_0$ $\Cc P^2_0$ is the mapping cylinder of the Hopf fibration $\eta:S^3\to S^2$ $\eta:S^3\to S^2$ . Recall that $S^6=S^2*S^3$ $S^6=S^2*S^3$ . Define an embedding $f:\Cc P^2_0\to S^6$ $f:\Cc P^2_0\to S^6$ by $f[(x,t)]:=[(x,\eta(x),t)]$ $f[(x,t)]:=[(x,\eta(x),t)]$ , where $x\in S^3$ $x\in S^3$ . In other words, the segment joining $x\in S^3$ $x\in S^3$ and $\eta(x)\in S^2$ $\eta(x)\in S^2$ is mapped onto the arc in

Tex syntax error

$S^6$ joining $x$ $x$ to $\eta(x)$ $\eta(x)$ . Clearly, the boundary 3-sphere of $\Cc P^2_0$ $\Cc P^2_0$ is standardly embedded into

Tex syntax error

$S^6$ . Hence $f$ $f$ extends to an embedding $\Cc P^2\to\Rr^7$ $\Cc P^2\to\Rr^7$ .

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 $\Rr^7$ ).

(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, $\S$ 2]) 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

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

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 $[N]$ .

Define `the Boechat-Haefliger invariant' of $f$

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

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

(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, $\S$ 2]. This is proved analogously to [Skopenkov2008, $\S$ 2, 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.

There is an isomorphism $E^7_D(S^4)\cong\Zz_{12}$ .

Let $N$ be a closed connected orientable 4-manifold.

(a) [Boechat&Haefliger1970] The image

Tex syntax error

${\rm im}\varkappa$ of the Boéchat-Haefliger invariant

$\displaystyle \varkappa:E^7_D(N)\to H_2(N)$ $\displaystyle \varkappa:E^7_D(N)\to H_2(N)$

$\displaystyle \text{is}\qquad \{u\in H_2(N)\ |\ u\equiv PDw_2(N)\mod2,\ u\cap u=\sigma(N)\}.$ $\displaystyle \text{is}\qquad \{u\in H_2(N)\ |\ u\equiv PDw_2(N)\mod2,\ u\cap u=\sigma(N)\}.$

(b) [Crowley&Skopenkov2008]

If $H_1(N)=0$ $H_1(N)=0$ , then for any

Tex syntax error

$u\in{\rm im}\varkappa$ there is an injective invariant called the Kreck invariant,

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

whose image is the subset of even elements.

Here

$PD:H^2(N;\Z_2)\to H_2(N;\Z_2)$ is the Poincaré duality isomorphism and $w_2(N)\in H^2(N;\Z_2)$ is the second Stiefel-Whitney class of $N$ , so that $PDw_2(N)$ is the second Stiefel-Whitney class of $N$ as defined by Stiefel as the obstruction to $N$ being spin.

$\cap:H_2(N)\times H_2(N)\to\Z$ is the intersection form.

$\sigma(N)$ is the signature (of the intersection form) of $N$ .

$\gcd(u,24)$ is the maximal integer $k$ such that both $u\in H_2(N)$ and 24 are divisible by $k$ .

$\eta_u$ is defined in [Crowley&Skopenkov2008].

Thus $\eta_u$ is surjective if $u$ is not divisible by 2.

Note that

Tex syntax error

$u\in{\rm im}\varkappa$ is divisible by 2 (for some $u$ $u$ or, equivalently, for any $u$ $u$ ) if and only if $N$ $N$ is spin.

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

(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

Tex syntax error

$\Zz^2$ are not in the image of $\varkappa$ $\varkappa$ .

Under the assumptions of Theorem 5.2 for any pair of embeddings $f:N\to\Rr^7$ and $g:S^4\to\Rr^7$

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

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.

(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.

6 References

[Boechat&Haefliger1970] J. Boéchat and A. Haefliger, Plongements différentiables des variétés orientées de dimension $4$ dans $\Rr^7$ , (French) Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York (1970), 156–166. MR0270384 (42 #5273) Zbl 0199.27003
[Boechat1971] J. Boéchat, Plongements de variétées différentiables orientées de dimension $4k$ dans $\Rr^{6k+1}$ , Comment. Math. Helv. 46 (1971), 141–161. MR0295373 (45 #4439) Zbl 0218.57016
[Crowley&Skopenkov2008] D. Crowley and A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, II, Intern. J. Math., 22:6 (2011) 731-757. Available at the arXiv:0808.1795.
[Crowley&Skopenkov2016] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, I. Classification modulo knots, Moscow Math. J., 21 (2021), 43--98. arXiv:1611.04738.
[Crowley&Skopenkov2016a] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, II. Smooth classification. Proc. A of the Royal Soc. of Edinburgh, to appear. arXiv:1612.04776
[Haefliger1962] A. Haefliger, Knotted $(4k-1)$ -spheres in $6k$ -space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
[Haefliger1966] A. Haefliger, Differential embeddings of $S^n$ in $S^{n+q}$ for $q>2$ , Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
[Haefliger1967] A. Haefliger, Lissage des immersions-I, Topology, 6 (1967) 221--240.
[Mandelbaum1980] R. Mandelbaum, Four-Dimensional Topology: An introduction, Bull. Amer. Math. Soc. (N.S.) 2 (1980) 1-159.
[Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
[Skopenkov2005] A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, Topol. Appl., 157 (2010) 2094-2110. Available at the arXiv:0512594.
[Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
[Skopenkov2008] A. Skopenkov, A classification of smooth embeddings of 3-manifolds in 6-space, Math. Z. 260 (2008), no.3, 647–672. Available at the arXiv:0603429 MR2434474 (2010e:57028) Zbl 1167.57013

[Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
[Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
[Skopenkov2016h] A. Skopenkov, High codimension links, to appear in Bull. Man. Atl.
[Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.

, $\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[$\S]{Skopenkov2016c}. {{beginthm|Remark|(PL and piecewise smooth embeddings)}}\label{r:plps} Any smooth manifold has a unique (up to PL homeomorphism) PL structure compatible with the given smooth structure \cite[Complement]{Milnor&Stasheff1974}. Since also any PL 4-manifold admits a unique smooth structure \cite[$\Sn-manifolds in $(2n-1)$ $(2n-1)$ -space which is discussed in [Skopenkov2016e], [Skopenkov2006, $\S$ $\S$ 2.4 `The Whitney invariant']. In this page we concentrate on more advanced results peculiar for $n=4$ $n=4$ .

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$ 1, $\S$ 3]. 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, $\S$ 5].

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, $\S$ 1.2], 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 normal framing on the standard embedding

Tex syntax error

${\rm i}:S^q\to S^m$ defines the standard embedding

Tex syntax error

${\rm i}_{m,q}:D^{m-q}\times S^q\to S^m$ . Denote by the same symbol

Tex syntax error

${\rm i}_{m,q}$ the restrictions of

Tex syntax error

${\rm i}_{m,q}$ to $S^p\times S^q$ $S^p\times S^q$ for $p\le m-q-1$ $p\le m-q-1$ .

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.

For an embedding $g:S^3\to D^6$ denote by $Sg$ the embedding

Tex syntax error

$\displaystyle S^1\times S^3\overset{g\times\id S^1}\to D^6\times S^1\overset{{\rm i}_{7,1}}\to S^7.$

The restriction of $Sg$ to $D^1_+\times S^3$ is isotopic to (the restriction to $D^1_+\times S^3$ of) 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.

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

Tex syntax error

$\displaystyle S^1\times S^3\overset{pr_2\times t^k}\to S^3\times S^3\overset{{\rm i}_{7,3}}\to S^7,$

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 $\eta:S^3\to S^2$ is the Hopf fibration and $S^2$ 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.

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

$\displaystyle S^1\times S^3 \xrightarrow{\overline\alpha\times pr_2} S^3\times S^3 \xrightarrow{{\rm i}_{7,3}} S^7.$

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 $\pi_3(V_{4, 2})=\Zz^2$ ).

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 $\tau(l,b)\#g$ for some $l,b\in\Z$ and $g\in E^7(S^4)$ .

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

$\displaystyle \nu^{-1}h(S^1)\cong S^1\times S^3\quad\text{and}\quad S^7-\nu^{-1}h(S^1)\sim \nu^{-1}h(S^2)\cong S^2\times S^3\not\sim S^2\vee S^3\vee S^5\sim S^7-{\rm i}_{7,3}(S^1\times S^3).$

The last homotopy equivalence is proved in a more general form

Tex syntax error

$S^m-{\rm i}_{m,q}(S^p\times S^q)\sim S^{m-p-1}\vee S^{m-q-1}\vee S^{m-p-q-1}$ for $m\ge p+q+3$ $m\ge p+q+3$ by induction on $p\ge0$ $p\ge0$ using the following observation: if $f:N\to S^n$ $f:N\to S^n$ is an embedding, then

Tex syntax error

$C_{{\rm i}\circ f}\sim\Sigma C_f$ .

We conjecture that $\nu^{-1}h(S^1)=\tau^1(S^1\times S^3)$ .

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

There is a smooth embedding $\Cc P^2\to\Rr^7$ .

Recall that $\Cc P^2_0$ $\Cc P^2_0$ is the mapping cylinder of the Hopf fibration $\eta:S^3\to S^2$ $\eta:S^3\to S^2$ . Recall that $S^6=S^2*S^3$ $S^6=S^2*S^3$ . Define an embedding $f:\Cc P^2_0\to S^6$ $f:\Cc P^2_0\to S^6$ by $f[(x,t)]:=[(x,\eta(x),t)]$ $f[(x,t)]:=[(x,\eta(x),t)]$ , where $x\in S^3$ $x\in S^3$ . In other words, the segment joining $x\in S^3$ $x\in S^3$ and $\eta(x)\in S^2$ $\eta(x)\in S^2$ is mapped onto the arc in

Tex syntax error

$S^6$ joining $x$ $x$ to $\eta(x)$ $\eta(x)$ . Clearly, the boundary 3-sphere of $\Cc P^2_0$ $\Cc P^2_0$ is standardly embedded into

Tex syntax error

$S^6$ . Hence $f$ $f$ extends to an embedding $\Cc P^2\to\Rr^7$ $\Cc P^2\to\Rr^7$ .

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 $\Rr^7$ ).

(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, $\S$ 2]) 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

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

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 $[N]$ .

Define `the Boechat-Haefliger invariant' of $f$

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

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

(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, $\S$ 2]. This is proved analogously to [Skopenkov2008, $\S$ 2, 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.

There is an isomorphism $E^7_D(S^4)\cong\Zz_{12}$ .

Let $N$ be a closed connected orientable 4-manifold.

(a) [Boechat&Haefliger1970] The image

Tex syntax error

${\rm im}\varkappa$ of the Boéchat-Haefliger invariant

$\displaystyle \varkappa:E^7_D(N)\to H_2(N)$ $\displaystyle \varkappa:E^7_D(N)\to H_2(N)$

$\displaystyle \text{is}\qquad \{u\in H_2(N)\ |\ u\equiv PDw_2(N)\mod2,\ u\cap u=\sigma(N)\}.$ $\displaystyle \text{is}\qquad \{u\in H_2(N)\ |\ u\equiv PDw_2(N)\mod2,\ u\cap u=\sigma(N)\}.$

(b) [Crowley&Skopenkov2008]

If $H_1(N)=0$ $H_1(N)=0$ , then for any

Tex syntax error

$u\in{\rm im}\varkappa$ there is an injective invariant called the Kreck invariant,

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

whose image is the subset of even elements.

Here

$PD:H^2(N;\Z_2)\to H_2(N;\Z_2)$ is the Poincaré duality isomorphism and $w_2(N)\in H^2(N;\Z_2)$ is the second Stiefel-Whitney class of $N$ , so that $PDw_2(N)$ is the second Stiefel-Whitney class of $N$ as defined by Stiefel as the obstruction to $N$ being spin.

$\cap:H_2(N)\times H_2(N)\to\Z$ is the intersection form.

$\sigma(N)$ is the signature (of the intersection form) of $N$ .

$\gcd(u,24)$ is the maximal integer $k$ such that both $u\in H_2(N)$ and 24 are divisible by $k$ .

$\eta_u$ is defined in [Crowley&Skopenkov2008].

Thus $\eta_u$ is surjective if $u$ is not divisible by 2.

Note that

Tex syntax error

$u\in{\rm im}\varkappa$ is divisible by 2 (for some $u$ $u$ or, equivalently, for any $u$ $u$ ) if and only if $N$ $N$ is spin.

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

(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

Tex syntax error

$\Zz^2$ are not in the image of $\varkappa$ $\varkappa$ .

Under the assumptions of Theorem 5.2 for any pair of embeddings $f:N\to\Rr^7$ and $g:S^4\to\Rr^7$

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

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.

(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.

6 References

[Boechat&Haefliger1970] J. Boéchat and A. Haefliger, Plongements différentiables des variétés orientées de dimension $4$ dans $\Rr^7$ , (French) Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York (1970), 156–166. MR0270384 (42 #5273) Zbl 0199.27003
[Boechat1971] J. Boéchat, Plongements de variétées différentiables orientées de dimension $4k$ dans $\Rr^{6k+1}$ , Comment. Math. Helv. 46 (1971), 141–161. MR0295373 (45 #4439) Zbl 0218.57016
[Crowley&Skopenkov2008] D. Crowley and A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, II, Intern. J. Math., 22:6 (2011) 731-757. Available at the arXiv:0808.1795.
[Crowley&Skopenkov2016] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, I. Classification modulo knots, Moscow Math. J., 21 (2021), 43--98. arXiv:1611.04738.
[Crowley&Skopenkov2016a] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, II. Smooth classification. Proc. A of the Royal Soc. of Edinburgh, to appear. arXiv:1612.04776
[Haefliger1962] A. Haefliger, Knotted $(4k-1)$ -spheres in $6k$ -space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
[Haefliger1966] A. Haefliger, Differential embeddings of $S^n$ in $S^{n+q}$ for $q>2$ , Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
[Haefliger1967] A. Haefliger, Lissage des immersions-I, Topology, 6 (1967) 221--240.
[Mandelbaum1980] R. Mandelbaum, Four-Dimensional Topology: An introduction, Bull. Amer. Math. Soc. (N.S.) 2 (1980) 1-159.
[Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
[Skopenkov2005] A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, Topol. Appl., 157 (2010) 2094-2110. Available at the arXiv:0512594.
[Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
[Skopenkov2008] A. Skopenkov, A classification of smooth embeddings of 3-manifolds in 6-space, Math. Z. 260 (2008), no.3, 647–672. Available at the arXiv:0603429 MR2434474 (2010e:57028) Zbl 1167.57013

[Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
[Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
[Skopenkov2016h] A. Skopenkov, High codimension links, to appear in Bull. Man. Atl.
[Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.

.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 normal framing on the standard embedding ${\rm i}:S^q\to S^m$ defines the standard embedding ${\rm i}_{m,q}:D^{m-q}\times S^q\to S^m$. Denote by the same symbol ${\rm i}_{m,q}$ the restrictions of ${\rm i}_{m,q}$ to $S^p\times S^q$ for $p\le m-q-1$. 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 [[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 $$S^1\times S^3\overset{g\times\id S^1}\to D^6\times S^1\overset{{\rm i}_{7,1}}\to S^7.$$ {{endthm}} The restriction of $Sg$ to $D^1_+\times S^3$ is isotopic to (the restriction to $D^1_+\times S^3$ of) 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 $$S^1\times S^3\overset{pr_2\times t^k}\to S^3\times S^3\overset{{\rm i}_{7,3}}\to S^7,$$ 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 $\eta:S^3\to S^2$ is the Hopf fibration and $S^2$ 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 $$S^1\times S^3 \xrightarrow{\overline\alpha\times pr_2} S^3\times S^3 \xrightarrow{{\rm i}_{7,3}} S^7.$$ 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 $\tau(l,b)\#g$ for some $l,b\in\Z$ and $g\in E^7(S^4)$. {{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 $$\nu^{-1}h(S^1)\cong S^1\times S^3\quad\text{and}\quad S^7-\nu^{-1}h(S^1)\sim \nu^{-1}h(S^2)\cong S^2\times S^3\not\sim S^2\vee S^3\vee S^5\sim S^7-{\rm i}_{7,3}(S^1\times S^3).$$ The last homotopy equivalence is proved in a more general form $S^m-{\rm i}_{m,q}(S^p\times S^q)\sim S^{m-p-1}\vee S^{m-q-1}\vee S^{m-p-q-1}$ for $m\ge p+q+3$ by induction on $p\ge0$ using the following observation: if $f:N\to S^n$ is an embedding, then $C_{{\rm i}\circ f}\sim\Sigma C_f$. We conjecture that $\nu^{-1}h(S^1)=\tau^1(S^1\times S^3)$. {{endthm}} {{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 the Hopf fibration $\eta$. 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' of $f$ $$ \varkappa(f):=A_{f,2}^{-1}\left(A_{f,4}[N]\cap A_{f,4}[N]\right)\in H_2(N). $$ Clearly, a map $\varkappa:E^7_D(N)\to H_2(N)$ is well-defined by $\varkappa([f]):=\varkappa(f)$. {{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}}\label{clth4} Let $N$ be a closed connected orientable 4-manifold. (a) \cite{Boechat&Haefliger1970} The image ${\rm im}\varkappa$ of the [[#The_Boechat-Haefliger_invariant|Boéchat-Haefliger invariant]] $$\varkappa:E^7_D(N)\to H_2(N)$$ $$\text{is}\qquad \{u\in H_2(N)\ |\ u\equiv PDw_2(N)\mod2,\ u\cap u=\sigma(N)\}.$$ (b) \cite{Crowley&Skopenkov2008} If $H_1(N)=0$, then for any $u\in{\rm im}\varkappa$ 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 * $PD:H^2(N;\Z_2)\to H_2(N;\Z_2)$ is the Poincaré duality isomorphism and $w_2(N)\in H^2(N;\Z_2)$ is the second Stiefel-Whitney class of $N$, so that $PDw_2(N)$ is the second Stiefel-Whitney class of $N$ as defined by Stiefel as the obstruction to $N$ being spin. * $\cap:H_2(N)\times H_2(N)\to\Z$ is the [[Intersection form|intersection form]]. * $\sigma(N)$ is the signature (of the intersection form) of $N$. * $\gcd(u,24)$ is the maximal integer $k$ such that both $u\in H_2(N)$ and 24 are divisible by $k$. * $\eta_u$ is defined in \cite{Crowley&Skopenkov2008}. Thus $\eta_u$ is surjective if $u$ is not divisible by 2. Note that $u\in{\rm im}\varkappa$ is divisible by 2 (for some $u$ or, equivalently, for any $u$) if and only if $N$ is spin. For a classification when $H_1(N)\ne0$ see \cite{Crowley&Skopenkov2016} and \cite{Crowley&Skopenkov2016a}. {{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}. == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]]n-manifolds in $(2n-1)$ $(2n-1)$ -space which is discussed in [Skopenkov2016e], [Skopenkov2006, $\S$ $\S$ 2.4 `The Whitney invariant']. In this page we concentrate on more advanced results peculiar for $n=4$ $n=4$ .

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$ 1, $\S$ 3]. 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, $\S$ 5].

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, $\S$ 1.2], 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 normal framing on the standard embedding

Tex syntax error

${\rm i}:S^q\to S^m$ defines the standard embedding

Tex syntax error

${\rm i}_{m,q}:D^{m-q}\times S^q\to S^m$ . Denote by the same symbol

Tex syntax error

${\rm i}_{m,q}$ the restrictions of

Tex syntax error

${\rm i}_{m,q}$ to $S^p\times S^q$ $S^p\times S^q$ for $p\le m-q-1$ $p\le m-q-1$ .

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.

For an embedding $g:S^3\to D^6$ denote by $Sg$ the embedding

Tex syntax error

$\displaystyle S^1\times S^3\overset{g\times\id S^1}\to D^6\times S^1\overset{{\rm i}_{7,1}}\to S^7.$

The restriction of $Sg$ to $D^1_+\times S^3$ is isotopic to (the restriction to $D^1_+\times S^3$ of) 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.

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

Tex syntax error

$\displaystyle S^1\times S^3\overset{pr_2\times t^k}\to S^3\times S^3\overset{{\rm i}_{7,3}}\to S^7,$

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 $\eta:S^3\to S^2$ is the Hopf fibration and $S^2$ 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.

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

$\displaystyle S^1\times S^3 \xrightarrow{\overline\alpha\times pr_2} S^3\times S^3 \xrightarrow{{\rm i}_{7,3}} S^7.$

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 $\pi_3(V_{4, 2})=\Zz^2$ ).

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 $\tau(l,b)\#g$ for some $l,b\in\Z$ and $g\in E^7(S^4)$ .

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

$\displaystyle \nu^{-1}h(S^1)\cong S^1\times S^3\quad\text{and}\quad S^7-\nu^{-1}h(S^1)\sim \nu^{-1}h(S^2)\cong S^2\times S^3\not\sim S^2\vee S^3\vee S^5\sim S^7-{\rm i}_{7,3}(S^1\times S^3).$

The last homotopy equivalence is proved in a more general form

Tex syntax error

$S^m-{\rm i}_{m,q}(S^p\times S^q)\sim S^{m-p-1}\vee S^{m-q-1}\vee S^{m-p-q-1}$ for $m\ge p+q+3$ $m\ge p+q+3$ by induction on $p\ge0$ $p\ge0$ using the following observation: if $f:N\to S^n$ $f:N\to S^n$ is an embedding, then

Tex syntax error

$C_{{\rm i}\circ f}\sim\Sigma C_f$ .

We conjecture that $\nu^{-1}h(S^1)=\tau^1(S^1\times S^3)$ .

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

There is a smooth embedding $\Cc P^2\to\Rr^7$ .

Recall that $\Cc P^2_0$ $\Cc P^2_0$ is the mapping cylinder of the Hopf fibration $\eta:S^3\to S^2$ $\eta:S^3\to S^2$ . Recall that $S^6=S^2*S^3$ $S^6=S^2*S^3$ . Define an embedding $f:\Cc P^2_0\to S^6$ $f:\Cc P^2_0\to S^6$ by $f[(x,t)]:=[(x,\eta(x),t)]$ $f[(x,t)]:=[(x,\eta(x),t)]$ , where $x\in S^3$ $x\in S^3$ . In other words, the segment joining $x\in S^3$ $x\in S^3$ and $\eta(x)\in S^2$ $\eta(x)\in S^2$ is mapped onto the arc in

Tex syntax error

$S^6$ joining $x$ $x$ to $\eta(x)$ $\eta(x)$ . Clearly, the boundary 3-sphere of $\Cc P^2_0$ $\Cc P^2_0$ is standardly embedded into

Tex syntax error

$S^6$ . Hence $f$ $f$ extends to an embedding $\Cc P^2\to\Rr^7$ $\Cc P^2\to\Rr^7$ .

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 $\Rr^7$ ).

(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, $\S$ 2]) 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

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

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 $[N]$ .

Define `the Boechat-Haefliger invariant' of $f$

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

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

(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, $\S$ 2]. This is proved analogously to [Skopenkov2008, $\S$ 2, 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.

There is an isomorphism $E^7_D(S^4)\cong\Zz_{12}$ .

Let $N$ be a closed connected orientable 4-manifold.

(a) [Boechat&Haefliger1970] The image

Tex syntax error

${\rm im}\varkappa$ of the Boéchat-Haefliger invariant

$\displaystyle \varkappa:E^7_D(N)\to H_2(N)$ $\displaystyle \varkappa:E^7_D(N)\to H_2(N)$

$\displaystyle \text{is}\qquad \{u\in H_2(N)\ |\ u\equiv PDw_2(N)\mod2,\ u\cap u=\sigma(N)\}.$ $\displaystyle \text{is}\qquad \{u\in H_2(N)\ |\ u\equiv PDw_2(N)\mod2,\ u\cap u=\sigma(N)\}.$

(b) [Crowley&Skopenkov2008]

If $H_1(N)=0$ $H_1(N)=0$ , then for any

Tex syntax error

$u\in{\rm im}\varkappa$ there is an injective invariant called the Kreck invariant,

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

whose image is the subset of even elements.

Here

$PD:H^2(N;\Z_2)\to H_2(N;\Z_2)$ is the Poincaré duality isomorphism and $w_2(N)\in H^2(N;\Z_2)$ is the second Stiefel-Whitney class of $N$ , so that $PDw_2(N)$ is the second Stiefel-Whitney class of $N$ as defined by Stiefel as the obstruction to $N$ being spin.

$\cap:H_2(N)\times H_2(N)\to\Z$ is the intersection form.

$\sigma(N)$ is the signature (of the intersection form) of $N$ .

$\gcd(u,24)$ is the maximal integer $k$ such that both $u\in H_2(N)$ and 24 are divisible by $k$ .

$\eta_u$ is defined in [Crowley&Skopenkov2008].

Thus $\eta_u$ is surjective if $u$ is not divisible by 2.

Note that

Tex syntax error

$u\in{\rm im}\varkappa$ is divisible by 2 (for some $u$ $u$ or, equivalently, for any $u$ $u$ ) if and only if $N$ $N$ is spin.

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

(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

Tex syntax error

$\Zz^2$ are not in the image of $\varkappa$ $\varkappa$ .

Under the assumptions of Theorem 5.2 for any pair of embeddings $f:N\to\Rr^7$ and $g:S^4\to\Rr^7$

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

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.

(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.

6 References

[Boechat&Haefliger1970] J. Boéchat and A. Haefliger, Plongements différentiables des variétés orientées de dimension $4$ dans $\Rr^7$ , (French) Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York (1970), 156–166. MR0270384 (42 #5273) Zbl 0199.27003
[Boechat1971] J. Boéchat, Plongements de variétées différentiables orientées de dimension $4k$ dans $\Rr^{6k+1}$ , Comment. Math. Helv. 46 (1971), 141–161. MR0295373 (45 #4439) Zbl 0218.57016
[Crowley&Skopenkov2008] D. Crowley and A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, II, Intern. J. Math., 22:6 (2011) 731-757. Available at the arXiv:0808.1795.
[Crowley&Skopenkov2016] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, I. Classification modulo knots, Moscow Math. J., 21 (2021), 43--98. arXiv:1611.04738.
[Crowley&Skopenkov2016a] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, II. Smooth classification. Proc. A of the Royal Soc. of Edinburgh, to appear. arXiv:1612.04776
[Haefliger1962] A. Haefliger, Knotted $(4k-1)$ -spheres in $6k$ -space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
[Haefliger1966] A. Haefliger, Differential embeddings of $S^n$ in $S^{n+q}$ for $q>2$ , Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
[Haefliger1967] A. Haefliger, Lissage des immersions-I, Topology, 6 (1967) 221--240.
[Mandelbaum1980] R. Mandelbaum, Four-Dimensional Topology: An introduction, Bull. Amer. Math. Soc. (N.S.) 2 (1980) 1-159.
[Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
[Skopenkov2005] A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, Topol. Appl., 157 (2010) 2094-2110. Available at the arXiv:0512594.
[Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
[Skopenkov2008] A. Skopenkov, A classification of smooth embeddings of 3-manifolds in 6-space, Math. Z. 260 (2008), no.3, 647–672. Available at the arXiv:0603429 MR2434474 (2010e:57028) Zbl 1167.57013

[Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
[Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
[Skopenkov2016h] A. Skopenkov, High codimension links, to appear in Bull. Man. Atl.
[Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.

4-manifolds in 7-space

Revision as of 11:26, 28 March 2019

Contents

1 Introduction

2 Examples of knotted tori

3 Embeddings of the complex projective plane

4 The Boechat-Haefliger invariant

5 Classification

6 References

2 Examples of knotted tori

3 Embeddings of the complex projective plane

4 The Boechat-Haefliger invariant

5 Classification

6 References

2 Examples of knotted tori

3 Embeddings of the complex projective plane

4 The Boechat-Haefliger invariant

5 Classification

6 References

Personal tools

Namespaces

Variants

Views

Actions

Search

Navigation

Interaction

Toolbox

@@ Line 109: / Line 109: @@
 There is a smooth embedding $\Cc P^2\to\Rr^7$.
-Recall that $\Cc P^2_0$ is the mapping cylinder of the Hopf fibration $\eta$. 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 $\Cc P^2\to\Rr^7$.
+Recall that $\Cc P^2_0$ is the mapping cylinder of the Hopf fibration $\eta:S^3\to S^2$. 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}}