4-manifolds in 7-space

Revision as of 09:24, 28 April 2016

1 Introduction

For notation and conventions see high codimension embeddings. Denote by ${{Stub}} == Introduction == <wikitex>; For notation and conventions see [[High_codimension_embeddings#Notation and conventions|high codimension embeddings]]. Denote by $\eta:S^3\to S^2$ is the Hopf map. </wikitex> == Examples == <wikitex>; The Hudson tori $\Hud_{7,4,2}:S^2\times S^2\to\Rr^7$ and $\Hud_{7,4,2}:S^1\times S^3\to\Rr^7$ are defined in [[Embeddings_just_below_the_stable_range:_classification#Hudson_tori|Remark 3.5.d]]. Analogously to [[Embeddings_just_below_the_stable_range:_classification#Action_by_linked_embedded_connected_sum|the case $m=2n$]] for an orientable 4-manifold $N$, an embedding $f_0:N\to\Rr^7$ and a class $a\in H_1(N)$ one can construct an embedding $f_a:N\to\Rr^7$. However, this embedding is no longer well-defined. We have $W(f_u,f_0)=u$ for [[Embeddings_just_below_the_stable_range#The Whitney invariant|the Whitney invariant]]. </wikitex> === Embeddings of CP² into R⁷ === <wikitex>; We follow \cite{Boechat&Haefliger1970}, p. 164. Recall that $\Cc P^2_0$ is the mapping cylinder of $\eta$. Recall that $S^6=S^2*S^3$. Define an embedding $f:\Cc P^2_0\to S^6$ by $f[(x,t)]:=[(x,\eta(x),t)]$, where $x\in S^3$. In other words, the segment joining $x\in S^3$ and $\eta(x)\in S^2$ is mapped onto the arc in $S^6$ 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 $\Cc P^2\to\Rr^7$. Apriori this extension need not be unique (because it can be changed by a connected sum with an embedding $g:S^4\to D^6$). Surprisingly, it is unique, and is the only embedding $\Cc P^2\to\Rr^7$ (up to isotopy and a hyperplane reflection of $\Rr^7$). {{beginthm|Theorem}}\label{cp2} *There are exactly two isotopy classes of embeddings $\Cc P^2\to\Rr^7$ (differing by a hyperplane reflection of $\Rr^7$). *For each embeddings $f:\Cc P^2\to\Rr^7$ and $g:S^4\to\Rr^7$ the embedding $f\#g$ is isotopic to $f$. {{endthm}} This follows by \cite{Skopenkov2005}, Triviality Theorem (a) or by [[Embeddings_of_4-manifolds_in_7-space#Classification|general classification]]. </wikitex> === The Lambrechts torus and the Hudson torus === <wikitex>; These two embeddings $\tau^1,\tau^2:S^1\times S^3\to\Rr^7$ are defined \cite{Skopenkov2006} as compositions $S^1\times S^3\overset{p_2\times t^i}\to S^3\times S^3\subset\Rr^7$, where $i=1,2$, $p_2$ is the projection onto the second factor, $\subset$ is the standard inclusion and maps $t^i:S^1\times S^3\to S^3$ are defined below. We shall see that $t^i|_{S^1\times y}$ are embeddings for each $y\in S^3$, hence $\tau^1$ and $\tau^2$ are embeddings. Define $t^1(s,y):=sy$, where $S^3$ is identified with the set of unit length quaternions and $S^1\subset S^3$ with the set of unit length complex numbers. Define $t^2(e^{i\theta},y):=\eta(y)\cos\theta+\sin\theta$, where $S^2$ is identified with the 2-sphere formed by unit length quaternions of the form $ai+bj+ck$. Note that $\tau^2$ is PL isotopic to the Hudson torus $\Hud_{7,4,1}$ defined in [[Embeddings_just_below_the_stable_range:_classification#Hudson_tori|Remark 3.5.d]]. Take the Hopf fibration $S^3\to S^7\overset\eta\to S^4$. Take the standard embeding $S^2\subset S^4$. Its complement has the homotopy type of $S^1$. Then $im\tau^1=\eta^{-1}(S^1)\cong S^1\times S^3\subset S^7$. This is the construction of Lambrechts motivated by the following property: $$S^7-im\tau^1\simeq \eta^{-1}(S^2)\cong S^2\times S^3\not\simeq S^2\vee S^3\vee S^5\simeq S^7-im f_0,$$ where $f_0:S^1\times S^3\to S^7$ is the standard embedding. </wikitex> === The Haefliger torus === <wikitex>; This is a PL embedding $S^2\times S^2\to\Rr^7$ which is (locally flat but) not PL isotopic to a smooth embedding \cite{Boechat&Haefliger1970}, p.165, \cite{Boechat1971}, 6.2. Take [[Knots,_i.e._embeddings_of_spheres#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 $S^2\times S^2-Int D^4\to\Rr^7_+$ (see \cite{Skopenkov2006}, Figure 3.7.a). These two extensions together form the Haefliger torus (see \cite{Skopenkov2006}, Figure 3.7.b). </wikitex> == The Boechat-Haefliger invariant == <wikitex>; Let $N$ be a closed connected orientable 4-manifold. Fix an orientation on $N$ and an orientation on $\Rr^7$. A ''homology Seifert surface'' $A_f$ for $f$ is the image of the fundamental class $[N]$ under the composition $H_4(N)\to H^2(C_f)\to H_5(C_f,\partial C_f)$ of the Alexander and Poincar\'e duality isomorphisms. (This composition is an inverse to the composition $H_5(C_f,\partial C_f)\to H_4(\partial C_f)\to H_4(N)$ of the boundary map $\partial$ and the projection $\nu_f$ of the normal bundle, cf. \cite{Skopenkov2008}, the Alexander Duality Lemma; this justifies the name `homology Seifert surface'.) Define $BH(f)$ to be the image of $A_f^2=A_f\cap A_f$ under the composition $H_3(C_f,\partial C_f)\to H^4(C_f)\to H_2(N)$ of the Poincaré and Alexander duality isomorphisms. (This composition has a direct geometric definition $\nu\partial$ as above.) This new definition is equivalent to the original one \cite{Boechat&Haefliger1970} by \cite{Crowley&Skopenkov2008}, Section Lemma 3.1. </wikitex> == Classification == <wikitex>; <!-- The results of this subsection are proved in \cite{Crowley&Skopenkov2008} unless other references are given. Let $N$ be a closed connected orientable 4-manifold.--> 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 $H_1(N)=0$. Here we work in the smooth category. {{beginthm|Theorem}}\label{hae4} $E^7(S^4)\cong\Zz_{12}$. \cite{Haefliger1966}, \cite{Skopenkov2005}, \cite{Crowley&Skopenkov2008}. {{endthm}} {{beginthm|Theorem}}\label{clth4} \cite{Crowley&Skopenkov2008} Let $N$ be a closed connected 4-manifold such that $H_1(N)=0$. There is the Boéchat-Haefliger invariant $$BH:E^7(N)\to H_2(N)$$ whose image is $$im(BH)=\{u\in H_2(N)\ |\ u\equiv PDw_2(N)\mod2,\ u\cap u=\sigma(N)\}.$$ For each $u\in im(BH)$ there is an injective invariant called the Kreck invariant, $$\eta_u:BH^{-1}(u)\to\Zz_{\gcd(u,24)}$$ 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 $u\in im(BH)$ is divisible by 2 (for some $u$ or, equivalently, for each $u$) if and only if $N$ is spin. For the definition of the Kreck invariant see \cite{Crowley&Skopenkov2008}. Theorem \ref{clth4} implies that * There are exactly twelve isotopy classes of embeddings $N\to\Rr^7$ if $N$ is an integral homology 4-sphere (cf. Theorem \ref{hae4}). * For each integer $u$ there are exactly $\gcd(u,12)$ isotopy classes of embeddings $f:S^2\times S^2\to\Rr^7$ with $BH(f)=(2u,0)$, and the same holds for those with $BH(f)=(0,2u)$. Other values of $\Zz^2$ are not in the image of $BH$. (We take the standard basis in $H_2(S^2\times S^2)$.) Theorem \ref{clth4} implies the following examples (first proved in \cite{Skopenkov2005}) of the triviality and the effectiveness of the connected sum action $E^7(S^4)\to E^7(N)$. * Let $N$ be a closed connected 4-manifold such that $H_1(N)=0$ and the signature $\sigma(N)$ of $N$ is not divisible by the square of an integer $s\ge2$. Then for each embeddings $f:N\to\Rr^7$ and $g:S^4\to\Rr^7$ the embedding $f\#g$ is isotopic to $f$ (in other words, $BH$ is injective). * If $N$ is a closed connected 4-manifold such that $H_1(N)=0$ and $f(N)\subset\Rr^6$ for an embedding $f:N\to\Rr^7$, then for each embedding $g:S^4\to\Rr^7$ the embedding $f\#g$ is not isotopic to $f$. The following can be obtained using \cite{Crowley&Skopenkov2008} (but not using \cite{Skopenkov2005}). * 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]]. If $u=6k\pm1$, then for each embedding $g:S^4\to\Rr^7$ the embedding $f_u\#g$ is isotopic to $f_u$. (For a general integer $u$ the number of isotopy classes of embeddings $f_u\#g$ is $\gcd(u,12)$.) Under assumptions of Theorem \ref{clth4} for each pair of embeddings $f:N\to\Rr^7$ and $g:S^4\to\Rr^7$ $$BH(f\#g)=BH(f)\quad\text{and}\quad\eta_{BH(f)}(f\#g)\equiv\eta_{BH(f)}(f)+\eta_0(g)\mod\gcd(BH(f),24).$$ </wikitex> == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]]\eta:S^3\to S^2$ is the Hopf map.

2 Examples

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

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

2.1 Embeddings of CP² into R⁷

We follow [Boechat&Haefliger1970], p. 164. Recall that $\Cc P^2_0$ is the mapping cylinder of $\eta$. Recall that $S^6=S^2*S^3$. Define an embedding $f:\Cc P^2_0\to S^6$ by $f[(x,t)]:=[(x,\eta(x),t)]$, where $x\in S^3$. In other words, the segment joining $x\in S^3$ and $\eta(x)\in S^2$ is mapped onto the arc in $S^6$ 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 $\Cc P^2\to\Rr^7$.

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

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

2.2 The Lambrechts torus and the Hudson torus

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

Define $t^1(s,y):=sy$, where $S^3$ is identified with the set of unit length quaternions and $S^1\subset S^3$ with the set of unit length complex numbers.

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

Note that $\tau^2$ is PL isotopic to the Hudson torus $\Hud_{7,4,1}$ defined in Remark 3.5.d.

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

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

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

2.3 The Haefliger torus

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

3 The Boechat-Haefliger invariant

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

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

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

4 Classification

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

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

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

whose image is

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

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

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

whose image is the subset of even elements.

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

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

Corollary 4.3. (a) There are exactly twelve isotopy classes of embeddings $N\to\Rr^7$ if $N$ is an integral homology 4-sphere (cf. Theorem 4.1).

(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 $BH(f)=(2u,0)$, and the same holds for those with $BH(f)=(0,2u)$. Other values of $\Zz^2$ are not in the image of $BH$. (We take the standard basis in $H_2(S^2\times S^2)$.)

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

(d) 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 $f$.

(e) 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. 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 $\gcd(u,12)$.)

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

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

Corollaries 4.3.cd on the triviality and the effectiveness of the connected sum action $E^7(S^4)\to E^7(N)$ were first proved in [Skopenkov2005].

5 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:0603429MR2434474 (2010e:57028) Zbl 1167.57013