6-manifolds: 2-connected
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== Introduction == | == Introduction == | ||
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− | Let $\mathcal{M}_6$ 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 $M$ (the notation is used to be consistent with [[6-manifolds: 1-connected]]). |
− | The classification $\mathcal{M}_6$ was one of Smale's first applications of the [[Wikipedia:h-cobordism_theorem|h-cobordism]] theorem {{cite|Smale1962a|Corollary 1.3}}. The is a precise 6-dimensional analogue of the classification of [[Surface|orientable surfaces]]: every 2-connected 6-manifold $M$ is diffeomorphic to a [[Wikipedia:Connected-sum|connected-sum]] | + | The classification $\mathcal{M}_6(0)$ was one of Smale's first applications of the [[Wikipedia:h-cobordism_theorem|h-cobordism]] theorem {{cite|Smale1962a|Corollary 1.3}}. The is a precise 6-dimensional analogue of the classification of [[Surface|orientable surfaces]]: every 2-connected 6-manifold $M$ is diffeomorphic to a [[Wikipedia:Connected-sum|connected-sum]] |
$$ M \cong \sharp_r(S^3 \times S^3)$$ | $$ M \cong \sharp_r(S^3 \times S^3)$$ | ||
where by definition $\sharp_0(S^3 \times S^3) = S^6$ and in general $r$ is determined by the formula for the [[Wikipedia:Euler characteristic|Euler characteristic]] of $M$ | where by definition $\sharp_0(S^3 \times S^3) = S^6$ and in general $r$ is determined by the formula for the [[Wikipedia:Euler characteristic|Euler characteristic]] of $M$ | ||
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{{endthm}} | {{endthm}} | ||
Hence if $\Nn$ denotes the natural numbers we obtain a bijection | Hence if $\Nn$ denotes the natural numbers we obtain a bijection | ||
− | $$ \mathcal{M}_6\equiv \Nn,~~~[M] \mapsto \frac{1}{2}b_3(M).$$ | + | $$ \mathcal{M}_6(0)\equiv \Nn,~~~[M] \mapsto \frac{1}{2}b_3(M).$$ |
</wikitex> | </wikitex> | ||
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{{beginthm|Theorem}} | {{beginthm|Theorem}} | ||
Every topological 2-connected 6-manifold admits a smooth structure which is unique up to diffoemorphism. In particular, there is a bijection | Every topological 2-connected 6-manifold admits a smooth structure which is unique up to diffoemorphism. In particular, there is a bijection | ||
− | $$ \mathcal{M}_6 \rightarrow \mathcal{M}^{\Top}_6.$$ | + | $$ \mathcal{M}_6(0) \rightarrow \mathcal{M}^{\Top}_6.$$ |
{{endthm}} | {{endthm}} | ||
Revision as of 12:00, 27 November 2010
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
Tex syntax errorbe the set of diffeomorphism classes of closed smooth simply-connected 2-connected 6-manifolds (the notation is used to be consistent with 6-manifolds: 1-connected). The classification
Tex syntax errorwas one of Smale's first applications of the h-cobordism theorem [Smale1962a, Corollary 1.3]. The is a precise 6-dimensional analogue of the classification of orientable surfaces: every 2-connected 6-manifold is 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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Tex syntax error, see 6-manifolds: 1-connected.
2 Construction and examples
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
.
3 Invariants
Tex syntax errorthen:
-
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, - the third Betti-number of is given by
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, - the Euler characteristic of is given by
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, - the intersection form of is isomorphic to the sum of b-copies of
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, the standard skew-symmetric hyperbolic form onTex syntax error
.
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 byTex syntax error.
Tex syntax errordenotes the natural numbers we obtain a bijection
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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. In particular, there is a bijection
Proof.
For any such manifold we haveTex syntax errorand 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