6-manifolds: 2-connected
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== Introduction == | == Introduction == | ||
<wikitex>; | <wikitex>; | ||
− | Let | + | 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]] 6-manifolds $M$. |
− | $$ M \cong \sharp_r S^3 \times S^3$$ | + | |
− | where by definition $\sharp_0 S^3 \times S^3 = S^6$ and in general $r$ is determined by the formula for the [[Euler characteristic]] of $M$ | + | The classification $\mathcal{M}_6(0)$ was one of Smale's first applications of the [h-cobordism] theorem {{cite|Smale1962a|Corollary 1.3}}. The classification, as for [[Surface|oriented surfaces]] is strikingly simple: every 2-connected 6-manifold $M$ is diffeomorphic to a [[connected-sum]] |
+ | $$ M \cong \sharp_r(S^3 \times S^3)$$ | ||
+ | where by definition $\sharp_0(S^3 \times S^3) = S^6$ and in general $r$ is determined by the formula for the [[Euler characteristic]] of $M$ | ||
$$ \chi(M) = 2 - 2r.$$ | $$ \chi(M) = 2 - 2r.$$ | ||
* For the more general case where $H_2(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]]. | * For the more general case where $H_2(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]]. |
Revision as of 16:40, 7 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 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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Tex syntax errorand in general is determined by the formula for the Euler characteristic of
- 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.
- , the -fold connected sum of .
3 Invariants
Suppose that is diffeomorphic to then:
- ,
- 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 r-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 and denotes the natural numbers we obtain a bijection
5 Further discussion
5.1 Topological 2-connected 6-manifolds
...
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