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* For the more general case where $F H_1(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]]. | * For the more general case where $F H_1(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]]. | ||
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== Introduction == | == Introduction == | ||
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Let $\mathcal{M}_6(0)$ be the set of diffeomorphism classes of [[wikipedia:Closed_manifold|closed]] [[wikipedia:Differentiable_manifold|smooth]] [[wikipedia:Simply-connected|simply-connected]] 2-connected 6-manifolds $M$. | Let $\mathcal{M}_6(0)$ be the set of diffeomorphism classes of [[wikipedia:Closed_manifold|closed]] [[wikipedia:Differentiable_manifold|smooth]] [[wikipedia:Simply-connected|simply-connected]] 2-connected 6-manifolds $M$. | ||
Revision as of 17:55, 17 June 2010
The sandbox is the page where you can experiment with the wiki syntax. Feel free to write nonsense or clear the page whenever you want. |
- For the more general case where , see 6-manifolds: 1-connected.
- For the more general case where
Tex syntax error
, see 6-manifolds: 1-connected.
- For the more general case where
Tex syntax error
, see 6-manifolds: 1-connected.
- For the more general case where
Tex syntax error
, see 6-manifolds: 1-connected.
- For the more general case where
Tex syntax error
, see 6-manifolds: 1-connected.
1 Introduction
Tex syntax errorLet be the set of diffeomorphism classes of closed smooth simply-connected 2-connected 6-manifolds
Tex syntax error. 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
Tex syntax erroris diffeomorphic to a connected-sum
where by definition and in general is determined by the formula for the Euler characteristic of
Tex syntax error
- For the more general case where , see 6-manifolds: 1-connected.
1 Construction and examples
The following gives a complete list of 2-connected 6-manifolds up to diffeomorphism:
- , the standard 6-sphere.
- , the -fold connected sum of .
2 Invariants
Suppose thatTex syntax erroris diffeomorphic to then:
- ,
- the third Betti-number of
Tex syntax error
is given by , - the Euler characteristic of
Tex syntax error
is given by , - the intersection form of
Tex syntax error
is isomorphic to the sum of b-copies of , the standard skew-symmetric hyperbolic form on .
3 Classification
Recall that the following theorem was stated in other words in the introduction:
Theorem 7.1 [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
4 Further discussion
4.1 Topological 2-connected 6-manifolds
Let be the set of homeomorphism classes of topological 2-connected 6-manifolds.
Theorem 9.1. 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 manifoldTex syntax errorwe have and so
Tex syntax erroris smoothable (see 6-manifolds: 1-connected). Any two homeomorphic manifolds have the same Euler Characteristic and so by Theorem 7.1 are diffeomorphic.
4.2 Mapping class groups
...
2 References
- [Smale1962a] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR0153022 (27 #2991) Zbl 0109.41103