6-manifolds: 2-connected
(→Invariants) |
(→Topological 2-connected 6-manifolds) |
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$$ \mathcal{M}_6(e) \rightarrow \mathcal{M}^{\Top}_6(e).$$ | $$ \mathcal{M}_6(e) \rightarrow \mathcal{M}^{\Top}_6(e).$$ | ||
{{endthm}} | {{endthm}} | ||
− | + | ||
+ | {{beginproof}} | ||
+ | For any such manifold $M$ we have $H^4(M; \Zz/2) \cong 0$ and so $M$ is smoothable (see [[6-manifolds: 1-connected#Smoothing theory|6-manifolds: 1-connected]]). Any two homeomorphic manifolds have the same Euler Characteristic and so by Theorem \ref{thm:classification} are diffeomorphic. | ||
+ | {{endproof}} | ||
</wikitex> | </wikitex> | ||
Revision as of 18:12, 10 June 2010
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
Let be the set of diffeomorphism classes of closed smooth simply-connected 2-connected 6-manifolds .
The classification was one of Smale's first applications of the h-cobordism theorem [Smale1962a, Corollary 1.3]. The classification, as for oriented surfaces is strikingly simple: every 2-connected 6-manifold is diffeomorphic to a connected-sum
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- For the more general case where , see 6-manifolds: 1-connected.
2 Construction and examples
The following gives a complete list of 2-connected 6-manifolds up to diffeomorphism:
- , the standard 6-sphere.
-
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, the -fold connected sum of .
3 Invariants
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- ,
- the third Betti-number of is given by ,
- the Euler characteristic of is given by ,
- the intersection form of is isomorphic to the sum of b-copies of , the standard skew-symmetric hyperbolic form on .
4 Classification
Recall that the following theorem was stated in other words in the introduction:
Theorem 4.1 [Smale1962a, Corolary 1.3]. The semi-group of 2-connected 6-manifolds is generated by .
Hence if denotes the natural numbers we obtain a bijection
5 Further discussion
5.1 Topological 2-connected 6-manifolds
Let be the set of homeomorphism classes of topological 2-connected 6-manifolds.
Theorem 5.1. Every topological 2-connected 6-manifold admits a smooth structure which is unique up to diffoemorphism. That is, there is a bijection
Proof. For any such manifold we have and so is smoothable (see 6-manifolds: 1-connected). Any two homeomorphic manifolds have the same Euler Characteristic and so by Theorem 4.1 are diffeomorphic.
5.2 Mapping class groups
...
References
- [Smale1962a] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR0153022 (27 #2991) Zbl 0109.41103