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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 00:03, 3 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
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.
- For the more general case where
Tex syntax error
, see 6-manifolds: 1-connected.
== Introduction ==
Tex syntax error
Tex syntax error
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
Tex syntax error
Tex syntax error
- For the more general case where
Tex syntax error
, 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
.
Contents |
1 Invariants
Tex syntax erroris diffeomorphic to
Tex syntax errorthen:
-
Tex syntax error
, - the third Betti-number of
Tex syntax error
is given byTex syntax error
, - the Euler characteristic of
Tex syntax error
is given byTex syntax error
, - the intersection form of
Tex syntax error
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 4.1 [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
2 Further discussion
2.1 Topological 2-connected 6-manifolds
Tex syntax errorbe 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
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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 4.1 are diffeomorphic.
Mapping class groups
...
3 References
- [Smale1962a] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR0153022 (27 #2991) Zbl 0109.41103