6-manifolds: 2-connected
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The semi-group of 2-connected 6-manifolds is generated by $S^3 \times S^3$. | The semi-group of 2-connected 6-manifolds is generated by $S^3 \times S^3$. | ||
{{endthm}} | {{endthm}} | ||
− | Hence if | + | Hence if $\Nn$ denotes the natural numbers we obtain a bijection |
$$ \mathcal{M}_6(0)\equiv \Nn,~~~[M] \mapsto \frac{1}{2}b_3(M).$$ | $$ \mathcal{M}_6(0)\equiv \Nn,~~~[M] \mapsto \frac{1}{2}b_3(M).$$ | ||
</wikitex> | </wikitex> |
Revision as of 17:12, 7 June 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 6-manifolds . 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 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
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- For the more general case where
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, 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
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, 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 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