4-manifolds in 7-space
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− | 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_0\to\ | + | 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 $f:\Cc P^2_0\to\Rr^7$. |
− | 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, in fact it is unique, and is essentially the only embedding $\Cc P^2_0\to\ | + | 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, in fact it is unique, and is essentially the only embedding $\Cc P^2_0\to\Rr^7$. |
Revision as of 12:41, 7 March 2010
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Contents |
1 Introduction
For notation and conventions see high codimension embeddings. Denote by is the Hopf map.
2 Examples
There is The Hudson torus .
Analogously to the case for an orientable 4-manifold , an embedding and a class one can construct an embedding . However, this embedding is no longer well-defined.
We have for the Whitney invariant (which is defined analogously to The Whitney invariant for .
2.1 Embeddings of CP2 into R7
Tex syntax errorjoining to . Clearly, the boundary 3-sphere of is standardly embedded into
Tex syntax error. Hence extends to an embedding .
Apriori this extension need not be unique (because it can be changed by a connected sum with an embedding ). Surprisingly, in fact it is unique, and is essentially the only embedding .
2.2 The Lambrechts torus and the Hudson torus
These two embeddings are defined [Skopenkov2006] as compositions , where , is the projection onto the second factor, is the standard inclusion and maps are defined below. We shall see that are embeddings for each , hence and are embeddings.
Define , where is identified with the set of unit length quaternions and with the set of unit length complex numbers.
Define , where is identified with the 2-sphere formed by unit length quaternions of the form .
Note that is PL isotopic to The Hudson torus .
Take the Hopf fibration . Take the standard embeding . Its complement has the homotopy type of . Then . This is the construction of Lambrechts motivated by the following property:
where is the standard embedding.
2.3 The Haefliger torus
This is a PL embedding 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 . Extend it to a conical embedding . By [Haefliger1962], the trefoil knot also extends to a smooth embedding (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 be a closed connected orientable 4-manifold. Denote by the closure of the complement in to a tubular neighborhood of .
Fix an orientation on and an orientation on . A for is the image of the fundamental class under the composition of the Alexander and Poincar\'e-Lefschetz duality isomorphisms. (This composition is an inverse to the composition of the boundary map and the normal bundle map , cf. [Skopenkov2008], the Alexander Duality Lemma; this justifies the name `homology Seifert surface'.)
Define to be the image of under the composition of the Poincar\'e-Lefschetz and Alexander duality isomorphisms. (This composition has a direct geometric definition 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 dans , (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 dans , 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 -spheres in -space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
- [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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