4-manifolds in 7-space

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

1 Introduction

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

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

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$$== Introduction == <wikitex>; This page is intended not only for specialists in embeddings, but also for mathematician from other areas who want to apply or to learn the theory of embeddings. Basic results on 4-manifolds in 7-space are particular cases of results on [[Embeddings_just_below_the_stable_range:_classification#Classification_just_below_the_stable_range|n-manifolds in (2n-1)-space]] for n=4 \cite{Skopenkov2016e}. In this page we concentrate on more advanced results peculiar for n=4. For a [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|general introduction to embeddings]] as well as the [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Notation and conventions|notation and conventions]] used on this page, we refer to \cite[$\S\S$1, $\S$$\S$3].

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

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

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

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

where

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$\rm{i}$ is the standard inclusion.

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

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

Example 2.2. Two sembeddings $\tau^1,\tau^2:S^1\times S^3\to S^7$$\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$$k=1,2$ and maps $t^k:S^1\times S^3\to S^3$$t^k:S^1\times S^3\to S^3$ are defined below. We shall see that $t^k|_{S^1\times y}$$t^k|_{S^1\times y}$ is an embedding for each $y\in S^3$$y\in S^3$ and $k=1,2$$k=1,2$, hence $\tau^1$$\tau^1$ and $\tau^2$$\tau^2$ are embeddings.

Define $t^1(s,y):=sy$$t^1(s,y):=sy$, where $S^3$$S^3$ is identified with the set of unit length quaternions and $S^1\subset S^3$$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$$t^2(e^{i\theta},y):=\eta(y)\cos\theta+\sin\theta$, where $S^2$$S^2$ is identified with the 2-sphere formed by unit length quaternions of the form $ai+bj+ck$$ai+bj+ck$.

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

Example 2.2 can be generalized as follows.

Example 2.3. Define a map $\tau \colon \Z^2 \to E^7(S^1 \times S^3).$$\tau \colon \Z^2 \to E^7(S^1 \times S^3).$ Take a smooth map $\alpha:S^3\to V_{4,2}$$\alpha:S^3\to V_{4,2}$. Assuming that $V_{4, 2}\subset (\R^4)^2$$V_{4, 2}\subset (\R^4)^2$, we have $\alpha(x) = (\alpha_1(x), \alpha_2(x))$$\alpha(x) = (\alpha_1(x), \alpha_2(x))$. Define the adjunction map $\R^2 \times S^3 \to \R^4$$\R^2 \times S^3 \to \R^4$ by $((s, t), x) \mapsto \alpha_1(x)s + \alpha_2(x)t$$((s, t), x) \mapsto \alpha_1(x)s + \alpha_2(x)t$. (Assuming that $V_{4, 2}\subset (\R^4)^{\R^2}$$V_{4, 2}\subset (\R^4)^{\R^2}$, this map is obtained from $\alpha$$\alpha$ by the exponential law.) Denote by $\overline\alpha:S^1\times S^3\to S^3$$\overline\alpha:S^1\times S^3\to S^3$ the restriction of the adjunction map. We define the embedding $\tau_\alpha$$\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$$\tau$ by $\tau(l, b):=[\tau_{\alpha}]$$\tau(l, b):=[\tau_{\alpha}]$, where $\alpha\colon S^3 \to V_{4, 2}$$\alpha\colon S^3 \to V_{4, 2}$ represents $(l, b) \in \pi_3(V_{4, 2})$$(l, b) \in \pi_3(V_{4, 2})$ (for the standard identification $\pi_3(V_{4, 2})=\Zz^2$$\pi_3(V_{4, 2})=\Zz^2$).

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

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

We conjecture that

• every PS embedding $S^1\times S^3\to S^7$$S^1\times S^3\to S^7$ is PS isotopic to $\tau(l,b)$$\tau(l,b)$ for some $l,b\in\Z$$l,b\in\Z$.

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

Example 2.4 (the Lambrechts torus). There is a smooth embedding $S^1\times S^3\to S^7$$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$$S^3\to S^7\overset{\nu}\to S^4$. Take the Hopf linking $h:S^1\sqcup S^2\to S^4$$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$$\nu^{-1}h(S^1)=im\tau^1$.

Example 2.5 (the Haefliger torus [Boechat&Haefliger1970, p.165], [Boechat1971, 6.2]). There is a PL embedding $S^2\times S^2\to S^7$$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$$S^3\to S^6$. Extend it to a conical embedding $D^4\to D^7_-$$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_+$$S^2\times S^2-Int D^4\to D^7_+$ [Skopenkov2006, Figure 3.7.a]. These two extensions together form the Haefliger torus [Skopenkov2006, Figure 3.7.b].

3 Embeddings of the complex projective plabe

Example 3.1 [Boechat&Haefliger1970, p.164]. There is a smooth embedding $\Cc P^2\to\Rr^7$$\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$$g:S^4\to D^6$). Surprisingly, it is unique, and in the smooth category is the only embedding $\Cc P^2\to\Rr^7$$\Cc P^2\to\Rr^7$ (up to isotopy and a hyperplane reflection of $\Rr^7$$\Rr^7$).

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

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

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

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

4 The Boechat-Haefliger invariant

Let $N$$N$ be a closed connected orientable 4-manifold and $f:N\to\Rr^7$$f:N\to\Rr^7$ an embedding. Fix an orientation on $N$$N$ and an orientation on $\Rr^7$$\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$$\partial$ and the projection $\nu_f$$\nu_f$ is an isomorphism, cf. [Skopenkov2008, the Alexander Duality Lemma]. The inverse $A_{f,s}$$A_{f,s}$ to this composition is the homology Alexander Duality isomorphism; it equals to the composition $H_s(N)\to H^{6-s}(C_f)\to H_{s+1}(C_f,\partial)$$H_s(N)\to H^{6-s}(C_f)\to H_{s+1}(C_f,\partial)$ of the cohomology Alexander and Poincaré duality isomorphisms.

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

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

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

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

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

5 Classification

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

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

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

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

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

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

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

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

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

Addendum 5.4. Under the assumptions of Theorem 5.2 for each pair of embeddings $f:N\to\Rr^7$$f:N\to\Rr^7$ and $g:S^4\to\Rr^7$$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)$$E^7_D(S^4)$ on $E^7_D(N)$$E^7_D(N)$ is trivial and where it is effective.

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

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

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

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

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

6 References

• [Boechat&Haefliger1970] J. Boéchat and A. Haefliger, Plongements différentiables des variétés orientées de dimension $4$$4$ dans $\Rr^7$$\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$$4k$ dans $\Rr^{6k+1}$$\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)$$(4k-1)$-spheres in $6k$$6k$-space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
• [Haefliger1966] A. Haefliger, Differential embeddings of $S^n$$S^n$ in $S^{n+q}$$S^{n+q}$ for $q>2$$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:0603429MR2434474 (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 $\Hud_{7,4,2}(a):S^2\times S^2\to S^7$ 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}. Denote by $\eta:S^3\to S^2$ the Hopf fibration and by $pr_k$ the projection onto the $k$-th factor of a Cartesian product. Define $D^m_+,D^m_-\subset S^m$ by the equations $x_1\ge0$ and $x_1\le0$, respectively. {{beginthm|Example|(Spinning construction)}}\label{artin} For an embedding $g:S^3\to D^6$ denote by $Sg$ the embedding $$S^1\times S^3\overset{\id S^1\times g}\to S^1\times D^6\overset{\rm{i}}\to S^7,$$ where $\rm{i}$ is the standard inclusion. {{endthm}} The restriction of $Sg$ to $D^1_+\times S^3$ is isotopic to the standard embedding. We conjecture that if $t:S^3\to D^6$ is [[3-manifolds_in_6-space#Examples|the Haefliger trefoil knot]], then $St$ is not smoothly isotopic to the connected sum of the standard embedding and any embedding $S^4\to S^7$. The following Examples \ref{quat} and \ref{tau} appear in \cite{Skopenkov2006} but could be known earlier. {{beginthm|Example}}\label{quat} Two sembeddings $\tau^1,\tau^2:S^1\times S^3\to S^7$ are defined as compositions $$S^1\times S^3\overset{pr_2\times t^k}\to S^3\times S^3\overset{\rm{i}}\to S^7,$$ where $k=1,2$ and maps $t^k:S^1\times S^3\to S^3$ are defined below. We shall see that $t^k|_{S^1\times y}$ is an embedding for each $y\in S^3$ and $k=1,2$, hence $\tau^1$ and $\tau^2$ are embeddings. Define $t^1(s,y):=sy$, where $S^3$ is identified with the set of unit length quaternions and $S^1\subset S^3$ with the set of unit length complex numbers. Define $t^2(e^{i\theta},y):=\eta(y)\cos\theta+\sin\theta$, where $S^2$ is identified with the 2-sphere formed by unit length quaternions of the form $ai+bj+ck$. {{endthm}} It would be interesting to know if $\tau^2$ is PL or smoothly isotopic to the Hudson torus $\Hud_{7,4,1}(1)$. Example \ref{quat} can be generalized as follows. {{beginthm|Example}}\label{tau} Define a map $\tau \colon \Z^2 \to E^7(S^1 \times S^3).$ Take a smooth map $\alpha:S^3\to V_{4,2}$. Assuming that $V_{4, 2}\subset (\R^4)^2$, we have $\alpha(x) = (\alpha_1(x), \alpha_2(x))$. Define the adjunction map $\R^2 \times S^3 \to \R^4$ by $((s, t), x) \mapsto \alpha_1(x)s + \alpha_2(x)t$. (Assuming that $V_{4, 2}\subset (\R^4)^{\R^2}$, this map is obtained from $\alpha$ by the exponential law.) Denote by $\overline\alpha:S^1\times S^3\to S^3$ the restriction of the adjunction map. We define the embedding $\tau_\alpha$ to be the composition $$S^1\times S^3 \xrightarrow{\overline\alpha\times pr_2} S^3\times S^3 \xrightarrow{\rm{i}} S^7.$$ We define the map $\tau$ by $\tau(l, b):=[\tau_{\alpha}]$, where $\alpha\colon S^3 \to V_{4, 2}$ represents $(l, b) \in \pi_3(V_{4, 2})$ (for the standard identification $\pi_3(V_{4, 2})=\Zz^2$). {{endthm}} Clearly, $\tau^1=\tau(1,0)$ and $\tau^2=\tau(0,1)$. See [[Knotted_tori#Examples|a generalization]] in \cite{Skopenkov2016k}. It would be interesting to know if $\tau(l,b)$ is smoothly or piecewise smoothly (PS) isotopic to $\tau(l,b+2l)$ for each $b,l\in\Zz$. We conjecture that * every PS embedding $S^1\times S^3\to S^7$ is PS isotopic to $\tau(l,b)$ for some $l,b\in\Z$. * every smooth embedding $S^1\times S^3\to S^7$ is smoothly isotopic to $\tau(l,b)\#g$ for some $l,b\in\Z$ and embedding $g:S^4\to S^7$. {{beginthm|Example|(the Lambrechts torus)}}\label{lamb} There is a smooth embedding $S^1\times S^3\to S^7$ whose complement is not homotopy equivalent to the complement of the standard embedding. I learned this simple construction from P. Lambrechts. Take the Hopf fibration $S^3\to S^7\overset{\nu}\to S^4$. Take [[High_codimension_links#Examples|the Hopf linking]] $h:S^1\sqcup S^2\to S^4$ \cite{Skopenkov2016h}. Then $$\nu^{-1}h(S^1)\cong S^1\times S^3\quad\text{and}\quad S^7-\nu^{-1}h(S^1)\sim \nu^{-1}h(S^2)\cong S^2\times S^3\not\sim S^2\vee S^3\vee S^5\sim S^7-im \rm{i}.$$ {{endthm}} We conjecture that $\nu^{-1}h(S^1)=im\tau^1$. {{beginthm|Example|(the Haefliger torus \cite[p.165]{Boechat&Haefliger1970}, \cite[6.2]{Boechat1971})}}\label{s2s2} There is a PL embedding $S^2\times S^2\to S^7$ which is (locally flat but) not PL 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 plabe== ; {{beginthm|Example|\cite[p.164]{Boechat&Haefliger1970}}}\label{ecp2} There is a smooth embedding $\Cc P^2\to\Rr^7$. Recall that $\Cc P^2_0$ is the mapping cylinder of $\eta$. Recall that $S^6=S^2*S^3$. Define an embedding $f:\Cc P^2_0\to S^6$ by $f[(x,t)]:=[(x,\eta(x),t)]$, where $x\in S^3$. In other words, the segment joining $x\in S^3$ and $\eta(x)\in S^2$ is mapped onto the arc in $S^6$ joining $x$ to $\eta(x)$. Clearly, the boundary 3-sphere of $\Cc P^2_0$ is standardly embedded into $S^6$. Hence $f$ extends to an embedding $\Cc P^2\to\Rr^7$. {{endthm}} Apriori the above extension need not be unique (because it can be changed by embedded connected sum with an embedding $g:S^4\to D^6$). Surprisingly, it is unique, and in the smooth category is the only embedding $\Cc P^2\to\Rr^7$ (up to isotopy and a hyperplane reflection of $\Rr^7$). {{beginthm|Theorem}}\label{cp2} (a) There are exactly two smooth isotopy classes of smooth embeddings $\Cc P^2\to\Rr^7$ (differing by 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 $E^7_{PL}(\Cc P^2)\to\Z$. {{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. {{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 == ; 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}. Here we work in the smooth category. {{beginthm|Theorem|(\cite{Haefliger1966}, see also \cite{Skopenkov2005}, \cite{Crowley&Skopenkov2008})}} \label{hae4} There is an isomorphism $E^7_D(S^4)\cong\Zz_{12}$. {{endthm}} {{beginthm|Theorem|(\cite{Crowley&Skopenkov2008})}}\label{clth4} Let $N$ be a closed connected 4-manifold such that $H_1(N)=0$. Then the image of the [[#The_Boechat-Haefliger_invariant|Boéchat-Haefliger invariant]] $$\varkappa:E^7_D(N)\to H_2(N)$$ $$\text{is}\qquad im \varkappa=\{u\in H_2(N)\ |\ u\equiv PDw_2(N)\mod2,\ u\cap u=\sigma(N)\}.$$ For each $u\in im \varkappa$ there is an injective invariant called the Kreck invariant, $$\eta_u:\varkappa^{-1}(u)\to\Zz_{\gcd(u,24)}$$ whose image is the subset of even elements. {{endthm}} Here $\gcd(u,24)$ is the maximal integer $k$ such that both $u\in H_2(N)$ and 24 are divisible by $k$. Thus $\eta_u$ is surjective if $u$ is not divisible by 2. Note that $u\in im \varkappa$ is divisible by 2 (for some $u$ or, equivalently, for each $u$) if and only if $N$ is spin. For the definition of the Kreck invariant see \cite{Crowley&Skopenkov2008}. {{beginthm|Corollary}}\label{coclth4} (a) There are exactly twelve isotopy classes of embeddings $N\to\Rr^7$ if $N$ is an integral homology 4-sphere (cf. Theorem \ref{hae4}). (b) Identify $H_2(S^2\times S^2) = \Z^2$ using the standard basis. For 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 $\varkappa$. {{endthm}} {{beginthm|Addendum}}\label{ad4} Under the 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 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}. If $u=6k\pm1$, then for each embedding $g:S^4\to\Rr^7$ the embedding $f_u\#g$ is isotopic to $f_u$. Moreover, for a general integer $u$ the number of isotopy classes of embeddings $f_u\#g$ is $\gcd(u,12)$. (b) Let $N$ be a closed connected 4-manifold such that $H_1(N)=0$ and the signature $\sigma(N)$ of $N$ is not divisible by the square of an integer $s\ge2$. Then for each pair of embeddings $f:N\to\Rr^7$ and $g:S^4\to\Rr^7$ the embedding $f\#g$ is isotopic to $f$; in other words, $\varkappa$ is injective. (c) If $N$ is a closed connected 4-manifold such that $H_1(N)=0$ and $f(N)\subset\Rr^6$ for an embedding $f:N\to\Rr^7$, then for every embedding $g:S^4\to\Rr^7$ the embedding $f\#g$ is not isotopic to $f$. {{endthm}} We remark that Corollary \ref{corclth4}(b) was first proved in \cite{Skopenkov2005} independently of Theorem \ref{clth4}. For classification when $H_1(N)\ne0$ see \cite{Crowley&Skopenkov2016} and \cite{Crowley&Skopenkov2016a}. == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]]\S1, $\S$$\S$3].

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

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

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

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

where

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$\rm{i}$ is the standard inclusion.

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

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

Example 2.2. Two sembeddings $\tau^1,\tau^2:S^1\times S^3\to S^7$$\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$$k=1,2$ and maps $t^k:S^1\times S^3\to S^3$$t^k:S^1\times S^3\to S^3$ are defined below. We shall see that $t^k|_{S^1\times y}$$t^k|_{S^1\times y}$ is an embedding for each $y\in S^3$$y\in S^3$ and $k=1,2$$k=1,2$, hence $\tau^1$$\tau^1$ and $\tau^2$$\tau^2$ are embeddings.

Define $t^1(s,y):=sy$$t^1(s,y):=sy$, where $S^3$$S^3$ is identified with the set of unit length quaternions and $S^1\subset S^3$$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$$t^2(e^{i\theta},y):=\eta(y)\cos\theta+\sin\theta$, where $S^2$$S^2$ is identified with the 2-sphere formed by unit length quaternions of the form $ai+bj+ck$$ai+bj+ck$.

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

Example 2.2 can be generalized as follows.

Example 2.3. Define a map $\tau \colon \Z^2 \to E^7(S^1 \times S^3).$$\tau \colon \Z^2 \to E^7(S^1 \times S^3).$ Take a smooth map $\alpha:S^3\to V_{4,2}$$\alpha:S^3\to V_{4,2}$. Assuming that $V_{4, 2}\subset (\R^4)^2$$V_{4, 2}\subset (\R^4)^2$, we have $\alpha(x) = (\alpha_1(x), \alpha_2(x))$$\alpha(x) = (\alpha_1(x), \alpha_2(x))$. Define the adjunction map $\R^2 \times S^3 \to \R^4$$\R^2 \times S^3 \to \R^4$ by $((s, t), x) \mapsto \alpha_1(x)s + \alpha_2(x)t$$((s, t), x) \mapsto \alpha_1(x)s + \alpha_2(x)t$. (Assuming that $V_{4, 2}\subset (\R^4)^{\R^2}$$V_{4, 2}\subset (\R^4)^{\R^2}$, this map is obtained from $\alpha$$\alpha$ by the exponential law.) Denote by $\overline\alpha:S^1\times S^3\to S^3$$\overline\alpha:S^1\times S^3\to S^3$ the restriction of the adjunction map. We define the embedding $\tau_\alpha$$\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$$\tau$ by $\tau(l, b):=[\tau_{\alpha}]$$\tau(l, b):=[\tau_{\alpha}]$, where $\alpha\colon S^3 \to V_{4, 2}$$\alpha\colon S^3 \to V_{4, 2}$ represents $(l, b) \in \pi_3(V_{4, 2})$$(l, b) \in \pi_3(V_{4, 2})$ (for the standard identification $\pi_3(V_{4, 2})=\Zz^2$$\pi_3(V_{4, 2})=\Zz^2$).

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

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

We conjecture that

• every PS embedding $S^1\times S^3\to S^7$$S^1\times S^3\to S^7$ is PS isotopic to $\tau(l,b)$$\tau(l,b)$ for some $l,b\in\Z$$l,b\in\Z$.

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

Example 2.4 (the Lambrechts torus). There is a smooth embedding $S^1\times S^3\to S^7$$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$$S^3\to S^7\overset{\nu}\to S^4$. Take the Hopf linking $h:S^1\sqcup S^2\to S^4$$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$$\nu^{-1}h(S^1)=im\tau^1$.

Example 2.5 (the Haefliger torus [Boechat&Haefliger1970, p.165], [Boechat1971, 6.2]). There is a PL embedding $S^2\times S^2\to S^7$$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$$S^3\to S^6$. Extend it to a conical embedding $D^4\to D^7_-$$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_+$$S^2\times S^2-Int D^4\to D^7_+$ [Skopenkov2006, Figure 3.7.a]. These two extensions together form the Haefliger torus [Skopenkov2006, Figure 3.7.b].

3 Embeddings of the complex projective plabe

Example 3.1 [Boechat&Haefliger1970, p.164]. There is a smooth embedding $\Cc P^2\to\Rr^7$$\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$$g:S^4\to D^6$). Surprisingly, it is unique, and in the smooth category is the only embedding $\Cc P^2\to\Rr^7$$\Cc P^2\to\Rr^7$ (up to isotopy and a hyperplane reflection of $\Rr^7$$\Rr^7$).

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

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

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

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

4 The Boechat-Haefliger invariant

Let $N$$N$ be a closed connected orientable 4-manifold and $f:N\to\Rr^7$$f:N\to\Rr^7$ an embedding. Fix an orientation on $N$$N$ and an orientation on $\Rr^7$$\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$$\partial$ and the projection $\nu_f$$\nu_f$ is an isomorphism, cf. [Skopenkov2008, the Alexander Duality Lemma]. The inverse $A_{f,s}$$A_{f,s}$ to this composition is the homology Alexander Duality isomorphism; it equals to the composition $H_s(N)\to H^{6-s}(C_f)\to H_{s+1}(C_f,\partial)$$H_s(N)\to H^{6-s}(C_f)\to H_{s+1}(C_f,\partial)$ of the cohomology Alexander and Poincaré duality isomorphisms.

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

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

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

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

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

5 Classification

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

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

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

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

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

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

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

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

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

Addendum 5.4. Under the assumptions of Theorem 5.2 for each pair of embeddings $f:N\to\Rr^7$$f:N\to\Rr^7$ and $g:S^4\to\Rr^7$$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)$$E^7_D(S^4)$ on $E^7_D(N)$$E^7_D(N)$ is trivial and where it is effective.

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

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

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

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

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

6 References

• [Boechat&Haefliger1970] J. Boéchat and A. Haefliger, Plongements différentiables des variétés orientées de dimension $4$$4$ dans $\Rr^7$$\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$$4k$ dans $\Rr^{6k+1}$$\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)$$(4k-1)$-spheres in $6k$$6k$-space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
• [Haefliger1966] A. Haefliger, Differential embeddings of $S^n$$S^n$ in $S^{n+q}$$S^{n+q}$ for $q>2$$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:0603429MR2434474 (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.