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== Further discussion == | == Further discussion == | ||
=== Topological 2-connected 6-manifolds === | === Topological 2-connected 6-manifolds === | ||
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Let $\mathcal{M}^{\Top}_6(e)$ be the set of homeomorphism classes of topological 2-connected 6-manifolds. | Let $\mathcal{M}^{\Top}_6(e)$ be the set of homeomorphism classes of topological 2-connected 6-manifolds. | ||
{{beginthm|Theorem}} | {{beginthm|Theorem}} | ||
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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. | 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}} | {{endproof}} | ||
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=== Mapping class groups === | === Mapping class groups === |
Revision as of 10:14, 2 July 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
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, see 6-manifolds: 1-connected.
- For the more general case where
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, see 6-manifolds: 1-connected.
- For the more general case where
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, see 6-manifolds: 1-connected.
- For the more general case where
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, see 6-manifolds: 1-connected.
- For the more general case where
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, see 6-manifolds: 1-connected.
Contents |
1 Introduction
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Tex syntax errorbe the set of diffeomorphism classes of closed smooth simply-connected 2-connected 6-manifolds
Tex syntax error. The classification
Tex syntax errorwas 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
Tex syntax error
Tex syntax errorand in general
Tex syntax erroris determined by the formula for the Euler characteristic of
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- For the more general case where
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, see 6-manifolds: 1-connected.
Construction and examples Tex syntax error
The following gives a complete list of 2-connected 6-manifolds up to diffeomorphism:
-
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, the standard 6-sphere. -
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, theTex syntax error
-fold connected sum ofTex syntax error
.
2 Invariants
Tex syntax erroris diffeomorphic to
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-
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, - the third Betti-number of
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is given byTex syntax error
, - the Euler characteristic of
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is given byTex syntax error
, - the intersection form of
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is isomorphic to the sum of b-copies ofTex syntax error
, the standard skew-symmetric hyperbolic form onTex syntax error
.
Classification
Recall that the following theorem was stated in other words in the introduction:
Theorem 5.1 [Smale1962a, Corolary 1.3].
The semi-group of 2-connected 6-manifolds is generated byTex syntax error.
Tex syntax errordenotes the natural numbers we obtain a bijection
3 Further discussion
3.1 Topological 2-connected 6-manifolds
Tex syntax errorbe the set of homeomorphism classes of topological 2-connected 6-manifolds.
Theorem 6.1. Every topological 2-connected 6-manifold admits a smooth structure which is unique up to diffoemorphism. That is, there is a bijection
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Proof.
For any such manifoldTex syntax errorwe have
Tex syntax errorand so
Tex syntax erroris smoothable (see 6-manifolds: 1-connected). Any two homeomorphic manifolds have the same Euler Characteristic and so by Theorem 5.1 are diffeomorphic.
Mapping class groups
...
4 References
- [Smale1962a] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR0153022 (27 #2991) Zbl 0109.41103