4-manifolds in 7-space

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1 Introduction

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

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

See general introduction on embeddings, notation and conventions in [Skopenkov2016c, ${{Stub}} == Introduction == <wikitex>; This page is intended not only for specialists in embeddings, but also for mathematician from other areas who want to apply or to learn the theory of embeddings. Basic results on 4-manifolds in 7-space are particular cases of results on [[Embeddings_just_below_the_stable_range:_classification|n-manifolds in (2n-1)-space]] for n=4 \cite{Skopenkov2016e}. In this page we concentrate on more advanced results peculiar for n=4. See [[High_codimension_embeddings#Introduction|general introduction on embeddings]], [[High_codimension_embeddings#Notation and conventions|notation and conventions]] in \cite[$\S\S$ 1, $\S$ 2].

2 Examples of knotted tori

There are the standard embeddings

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$\rm{{i}}:S^p\times S^q\to S^7$ for $p,q>0$ $p,q>0$ , $p+q\le6$ $p+q\le6$ (which are defined in [Skopenkov2015a, $\S$ $\S$ 2.1]).

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

Denote by $\eta:S^3\to S^2$ the Hopf fibration and by $pr_k$ the projection onto the $k$ -th factor of a Cartesian product.

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

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

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 $ai+bj+ck$ .

These examples appear in [Skopenkov2006] but could be known earlier. Note that $\tau^2$ is PL isotopic to the Hudson torus $\Hud_{7,4,1}$ defined in Remark 3.5.d of [Skopenkov2016e].

Example 2.1 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

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

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

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

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

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

There is a PL embedding $S^2\times S^2\to S^7$ which is (locally flat but) not PL isotopic to a smooth embedding.

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

3 Other examples

For an orientable 4-manifold $N$ , an embedding $f_0:N\to\Rr^7$ and a class $a\in H_2(N)$ one can construct an embedding $f_a:N\to\Rr^7$ by linked connected sum analogously to embeddings into $\Rr^8$ [Skopenkov2016e]. If $N$ is simply-connected, this gives a free transitive action of $H_2(N)$ on $E^7_{PL}(N)$ [Skopenkov2016e]. We also have $W(f_a,f_0)=a$ for the Whitney invariant which is defined in [Skopenkov2016e].

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

Recall that $\Cc P^2_0$ $\Cc P^2_0$ is the mapping cylinder of $\eta$ $\eta$ . 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

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$S^6$ 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

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$S^6$ . 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 is the only embedding $\Cc P^2\to\Rr^7$ (up to isotopy and a hyperplane reflection of $\Rr^7$ ).

Theorem 3.2.

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

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

(c) The Whitney invariant is a 1--1 correspondence $E^7_{PL}(\Cc P^2)\to\Z$ . The inverse is defined using linked connected sum [Skopenkov2016e].

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

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 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 Poincar\'e 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

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

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

(b) We have $\varkappa(f)-\varkappa(f_0)=\pm2W(f,f_0)$ for the Whitney invariant $W(f,f_0)$ [Skopenkov2016e]. This is proved analogously to [Skopenkov2008, $\S$ 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 $\varkappa(f)\mod2=PDw_2(N)$ .

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

5 Classification

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

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

Let $N$ be a closed connected 4-manifold such that $H_1(N)=0$ . Then the image of the Boéchat-Haefliger invariant

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

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

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

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

whose image is the subset of even elements.

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

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

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

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

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

The following corollaries are examples of the effectiveness and the triviality of the embedded connected sum action of $E^7_D(S^4)$ on $E^7_D(N)$ .

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

(b) Let $N$ be a closed connected 4-manifold such that $H_1(N)=0$ and the signature $\sigma(N)$ of $N$ is not divisible by the square of an integer $s\ge2$ . Then for each 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. (First proved in [Skopenkov2005] independently of Theorem 5.2.)

(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 each embedding $g:S^4\to\Rr^7$ the embedding $f\#g$ is not isotopic to $f$ .

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.
[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
[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

[Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470

[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}. == Examples of knotted tori== ; There are the standard embeddings $\rm{{i}}:S^p\times S^q\to S^7$ for $p,q>0$, $p+q\le6$ (which are defined in \cite[$\S.1]{Skopenkov2015a}). The Hudson tori $\Hud_{7,4,2}:S^2\times S^2\to\Rr^7$ and $\Hud_{7,4,1}:S^1\times S^3\to\Rr^7$ are defined in [[Embeddings_just_below_the_stable_range:_classification#Hudson_tori|Remark 3.5.d]] of \cite{Skopenkov2016e}. Denote by $\eta:S^3\to S^2$ the Hopf fibration and by $pr_k$ the projection onto the $k$-th factor of a Cartesian product. {{beginthm|Example}}\label{quat} Two embeddings $\tau^1,\tau^2:S^1\times S^3\to\Rr^7$ are defined as compositions $$S^1\times S^3\overset{pr_2\times t^k}\to S^3\times S^3\overset{\rm{i}}\to\Rr^7,$$ 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}} These examples appear in \cite{Skopenkov2006} but could be known earlier. Note that $\tau^2$ is PL isotopic to the Hudson torus $\Hud_{7,4,1}$ defined in [[Embeddings_just_below_the_stable_range:_classification#Hudson_tori|Remark 3.5.d]] of \cite{Skopenkov2016e}. 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 $\pi_3(V_{4, 2})=\Z^2$). {{endthm}} Clearly, $\tau^1=\tau(1,0)$ and $\tau^2=\tau(0,1)$. See [[Knotted_tori#Examples|a generalization]] in \cite{Skopenkov2016k}. {{beginthm|Example}}\label{lamb} There is an 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 [[Links,_i.e._embeddings_of_non-connected_manifolds#The_Hopf_linking|the Hopf linking]] $h:S^1\sqcup S^2\to S^4$ \cite{Skopenkov2016h}. Then $$\nu^{-1}h(S^1)\cong S^1\times S^3\quad\text{and}\quad S^7-\nu^{-1}h(S^1)\sim \nu^{-1}h(S^2)\cong S^2\times S^3\not\sim S^2\vee S^3\vee S^5\sim S^7-im \rm{i}.$$ {{endthm}} We conjecture that $\nu^{-1}h(S^1)=im\tau^1$. {{beginthm|Example|(\cite[p.165]{Boechat&Haefliger1970}, \cite[6.2]{Boechat1971})}}\label{s2s2} There is a PL embedding $S^2\times S^2\to\Rr^7$ which is (locally flat but) not PL isotopic to a smooth embedding. Take [[3-manifolds_in_6-space#The_Haefliger_trefoil_knot|the Haefliger trefoil knot]] $S^3\to\Rr^6$. Extend it to a conical embedding $D^4\to\Rr^7_-$. By \cite{Haefliger1962}, the trefoil knot also extends to a smooth embedding $S^2\times S^2-Int D^4\to\Rr^7_+$ \cite[Figure 3.7.a]{Skopenkov2006}. These two extensions together form the Haefliger torus \cite[Figure 3.7.b]{Skopenkov2006}. {{endthm}} == Other examples == ; For an orientable 4-manifold $N$, an embedding $f_0:N\to\Rr^7$ and a class $a\in H_2(N)$ one can construct an embedding $f_a:N\to\Rr^7$ by linked connected sum analogously to [[Embeddings_just_below_the_stable_range:_classification#Action_by_linked_embedded_connected_sum|embeddings into $\Rr^8$]] \cite{Skopenkov2016e}. If $N$ is simply-connected, this gives a [[Embeddings_just_below_the_stable_range:_classification#A_generalization_to_highly-connected_manifolds|free transitive action]] of $H_2(N)$ on $E^7_{PL}(N)$ \cite{Skopenkov2016e}. We also have $W(f_a,f_0)=a$ for [[Embeddings_just_below_the_stable_range#The Whitney invariant|the Whitney invariant]] which is defined in \cite{Skopenkov2016e}. {{beginthm|Example|(\cite[p.164]{Boechat&Haefliger1970}}}\label{ecp2} There is an 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 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 a hyperplane reflection of $\Rr^7$). (b) For each pair of smooth embeddings $f:\Cc P^2\to\Rr^7$ and $g:S^4\to\Rr^7$ the embedding $f\#g$ is smoothly isotopic to $f$. (c) [[Embeddings_just_below_the_stable_range:_classification#The_Whitney_invariant|The Whitney invariant]] is a 1--1 correspondence $E^7_{PL}(\Cc P^2)\to\Z$. The inverse is defined using [[Embeddings_just_below_the_stable_range:_classification#A_generalization_to_highly-connected_manifolds|linked connected sum]] \cite{Skopenkov2016e}. {{endthm}} Part (c) follows by \cite{Boechat&Haefliger1970}. Parts (a) and (b) are proved in \cite[Triviality Theorem (a)]{Skopenkov2005} or follow by [[#Classification|Theorem \ref{clth4}]] below. == 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 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 Poincar\'e duality isomorphisms. {{endthm}} {{beginthm|Definition}}\label{dbh} 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 $$ \varkappa(f):=A_{f,2}^{-1}\left(A_{f,4}[N]\cap A_{f,4}[N]\right)\in H_2(N). $$ {{endthm}} {{beginthm|Remark}} (a) Take a small oriented disk $D^3_f\subset\Rr^7$ whose intersection with $fN$ consists of exactly one point of sign $+1$ and such that $\partial D^3_f\subset\partial C_f$. A ''meridian of $f$'' is $\partial D^3_f$. A homology Seifert surface $Y\in H_5(C_f,\partial)$ for $f$ is uniquely defined by the condition $Y\cap [\partial D^3_f]=1$. (b) We have $\varkappa(f)-\varkappa(f_0)=\pm2W(f,f_0)$ for [[Embeddings_just_below_the_stable_range#The Whitney invariant|the Whitney invariant]] $W(f,f_0)$ \cite{Skopenkov2016e}. 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 $\varkappa(f)\mod2=PDw_2(N)$. (d) Earlier notation was $w_f$ \cite{Boechat&Haefliger1970}, $BH(f)$ \cite{Skopenkov2005} and $\aleph(f)$ \cite{Crowley&Skopenkov2008}. {{endthm}} == Classification == ; See [[Embeddings_just_below_the_stable_range:_classification#Classification_just_below_the_stable_range|a classification]] of $E^7_{PL}(N)$ for a closed connected 4-manifold $N$ such that $H_1(N)=0$ in \cite{Skopenkov2016e}. Here we work in the smooth category. {{beginthm|Theorem|(\cite{Haefliger1966}, see also \cite{Skopenkov2005}, \cite{Crowley&Skopenkov2008})}}\label{hae4} $E^7_D(S^4)\cong\Zz_{12}$. {{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]] $$ \varkappa:E^7_D(N)\to H_2(N) $$ $$ \text{is}\quad im \varkappa=\{u\in H_2(N)\ |\ u\equiv PDw_2(N)\mod2,\ u\cap u=\sigma(N)\}. $$ 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 $\gcd(u,24)$ 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) For each integer $u$ there are exactly $\gcd(u,12)$ isotopy classes of embeddings $f:S^2\times S^2\to\Rr^7$ with $\varkappa(f)=(2u,0)$, and the same holds for those with $\varkappa(f)=(0,2u)$. Other values of $\Zz^2$ are not in the image of $\varkappa$. (We take the standard basis in $H_2(S^2\times S^2)$.) {{endthm}} {{beginthm|Addendum}}\label{ad4} Under assumptions of Theorem \ref{clth4} for each 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 corollaries are examples of the effectiveness and the triviality of the embedded connected sum action of $E^7_D(S^4)$ on $E^7_D(N)$. {{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}. If $u=6k\pm1$, then for each embedding $g:S^4\to\Rr^7$ the embedding $f_u\#g$ is isotopic to $f_u$. (For a general integer $u$ the number of isotopy classes of embeddings $f_u\#g$ is $\gcd(u,12)$.) (b) Let $N$ be a closed connected 4-manifold such that $H_1(N)=0$ and the signature $\sigma(N)$ of $N$ is not divisible by the square of an integer $s\ge2$. Then for each 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. (First proved in \cite{Skopenkov2005} independently of Theorem \ref{clth4}.) (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 each embedding $g:S^4\to\Rr^7$ the embedding $f\#g$ is not isotopic to $f$. {{endthm}} == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]]\S1, $\S$ $\S$ 2].

2 Examples of knotted tori

There are the standard embeddings

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$\rm{{i}}:S^p\times S^q\to S^7$ for $p,q>0$ $p,q>0$ , $p+q\le6$ $p+q\le6$ (which are defined in [Skopenkov2015a, $\S$ $\S$ 2.1]).

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

Denote by $\eta:S^3\to S^2$ the Hopf fibration and by $pr_k$ the projection onto the $k$ -th factor of a Cartesian product.

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

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

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 $ai+bj+ck$ .

These examples appear in [Skopenkov2006] but could be known earlier. Note that $\tau^2$ is PL isotopic to the Hudson torus $\Hud_{7,4,1}$ defined in Remark 3.5.d of [Skopenkov2016e].

Example 2.1 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

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

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

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

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

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

There is a PL embedding $S^2\times S^2\to S^7$ which is (locally flat but) not PL isotopic to a smooth embedding.

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

3 Other examples

For an orientable 4-manifold $N$ , an embedding $f_0:N\to\Rr^7$ and a class $a\in H_2(N)$ one can construct an embedding $f_a:N\to\Rr^7$ by linked connected sum analogously to embeddings into $\Rr^8$ [Skopenkov2016e]. If $N$ is simply-connected, this gives a free transitive action of $H_2(N)$ on $E^7_{PL}(N)$ [Skopenkov2016e]. We also have $W(f_a,f_0)=a$ for the Whitney invariant which is defined in [Skopenkov2016e].

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

Recall that $\Cc P^2_0$ $\Cc P^2_0$ is the mapping cylinder of $\eta$ $\eta$ . 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

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$S^6$ 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

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$S^6$ . 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 is the only embedding $\Cc P^2\to\Rr^7$ (up to isotopy and a hyperplane reflection of $\Rr^7$ ).

Theorem 3.2.

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

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

(c) The Whitney invariant is a 1--1 correspondence $E^7_{PL}(\Cc P^2)\to\Z$ . The inverse is defined using linked connected sum [Skopenkov2016e].

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

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 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 Poincar\'e 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

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

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

(b) We have $\varkappa(f)-\varkappa(f_0)=\pm2W(f,f_0)$ for the Whitney invariant $W(f,f_0)$ [Skopenkov2016e]. This is proved analogously to [Skopenkov2008, $\S$ 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 $\varkappa(f)\mod2=PDw_2(N)$ .

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

5 Classification