6-manifolds: 2-connected
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* the third Betti-number of $M$ is given by $b_3(M) = 2r$, | * the third Betti-number of $M$ is given by $b_3(M) = 2r$, | ||
* the Euler characteristic of $M$ is given by $\chi(M) = 2 = 2r$, | * the Euler characteristic of $M$ is given by $\chi(M) = 2 = 2r$, | ||
− | * the [[intersection form]] of $M$ is isomorphic to the sum of r-copies of $H_{-}(\Zz)$, the standard skew-symmetric hyperbolic form on $\Zz^2$. | + | * the [[Intersection forms|intersection form]] of $M$ is isomorphic to the sum of r-copies of $H_{-}(\Zz)$, the standard skew-symmetric hyperbolic form on $\Zz^2$. |
</wikitex> | </wikitex> | ||
Revision as of 18:08, 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
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.
2 Construction and examples
The following gives a complete list of 2-connected 6-manifolds up to diffeomorphism:
-
Tex syntax error
, the standard 6-sphere. - , the
Tex syntax error
-fold connected sum ofTex syntax error
.
3 Invariants
Tex syntax erroris diffeomorphic to then:
- ,
- the third Betti-number of
Tex syntax error
is given by , - the Euler characteristic of
Tex syntax error
is given by , - the intersection form of
Tex syntax error
is isomorphic to the sum of r-copies ofTex syntax error
, 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.
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