4-manifolds in 7-space

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1 Introduction

For notation and conventions see high codimension embeddings. Denote by $\eta:S^3\to S^2$$== 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>; There is [[Embeddings_just_below_the_stable_range#Examples#Remarks|The Hudson torus]] $\Hud_{7,4,2}:S^2\times S^2\to\Rr^7$. Analogously to [[Embeddings_just_below_the_stable_range#Examples#An action of the first homology group on embeddings|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 (which is defined analogously to [[Embeddings_just_below_the_stable_range#The Whitney invariant (for either n odd or N orientable)|The Whitney invariant for $m=2n$]]. </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 $f:\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). </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\rightarrow 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 [[Embeddings_just_below_the_stable_range#Examples#Remarks|The Hudson torus]] $\Hud_{7,4,1}$. 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{Haefliger1962}, \cite{Boechat&Haefliger1970}, p.165, \cite{Boechat1971}, 6.2. Take [[Embeddings_of_3-manifolds_in_6-space#Examples#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}, 3.7.b). </wikitex> == The Boechat-Haefliger invariant == <wikitex>; Let $N$ be a closed connected orientable 4-manifold. Denote by $C_f$ the closure of the complement in $S^7\supset\Rr^7$ to a tubular neighborhood of $f(N)$. 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 $H_4(N)\to H^2(C_f)\to H_5(C_f,\partial)$ of the Alexander and Poincar\'e-Lefschetz duality isomorphisms. (This composition is an inverse to the composition $H_5(C_f,\partial)\to H_4(\partial C_f)\to H_4(N)$ of the boundary map $\partial$ and the normal bundle map $\nu$, 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 $H_3(C_f,\partial)\to H^4(C_f)\to H_2(N)$ of the Poincar\'e-Lefschetz 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. </wikitex> == Classification == <wikitex>; The results of this subsection are proved in \cite{Crowley&Skopenkov2008} unless other references are given. Let be a closed connected orientable 4-manifold. </wikitex> == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]] {{Stub}}\eta:S^3\to S^2$ is the Hopf map.

2 Examples

There is The Hudson torus $\Hud_{7,4,2}:S^2\times S^2\to\Rr^7$$\Hud_{7,4,2}:S^2\times S^2\to\Rr^7$.

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

We have $W(f_u,f_0)=u$$W(f_u,f_0)=u$ for the Whitney invariant (which is defined analogously to The Whitney invariant for $m=2n$$m=2n$.

2.1 Embeddings of CP2 into R7

We follow [Boechat&Haefliger1970], p. 164. 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 $S^6$$S^6$ joining $x$$x$ to $\eta(x)$$\eta(x)$. Clearly, the boundary 3-sphere of $\Cc P^2_0$$\Cc P^2_0$ is standardly embedded into $S^6$$S^6$. Hence $f$$f$ extends to an embedding $f:\Cc P^2\to\Rr^7$$f:\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$$g:S^4\to D^6$). Surprisingly, it is unique, and is the only embedding $\Cc P^2\to\Rr^7$$\Cc P^2\to\Rr^7$ (up to isotopy and a hyperplane reflection of $\Rr^7$$\Rr^7$).

Theorem 2.1.

• There are exactly two isotopy classes of embeddings $\Cc P^2\to\Rr^7$$\Cc P^2\to\Rr^7$ (differing by a hyperplane reflection of $\Rr^7$$\Rr^7$).
• For each 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$$f\#g$ is isotopic to $f$$f$.

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

2.2 The Lambrechts torus and the Hudson torus

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

Note that $\tau^2$$\tau^2$ is PL isotopic to The Hudson torus $\Hud_{7,4,1}$$\Hud_{7,4,1}$.

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

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

where $f_0:S^1\times S^3\to S^7$$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$$S^2\times S^2\to\Rr^7$ which is (locally flat but) not PL isotopic to a smooth embedding [Haefliger1962], [Boechat&Haefliger1970], p.165, [Boechat1971], 6.2. Take the Haefliger trefoil knot $S^3\to\Rr^6$$S^3\to\Rr^6$. Extend it to a conical embedding $D^4\to\Rr^7_-$$D^4\to\Rr^7_-$. By [Haefliger1962], the trefoil knot also extends to a smooth embedding $S^2\times S^2-Int D^4\to\Rr^7_+$$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], 3.7.b).

3 The Boechat-Haefliger invariant

Let $N$$N$ be a closed connected orientable 4-manifold. Denote by $C_f$$C_f$ the closure of the complement in $S^7\supset\Rr^7$$S^7\supset\Rr^7$ to a tubular neighborhood of $f(N)$$f(N)$.

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

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

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

4 Classification

The results of this subsection are proved in [Crowley&Skopenkov2008] unless other references are given. Let be a closed connected orientable 4-manifold.

5 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.
• [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
• [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

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