6-manifolds: 2-connected
(Difference between revisions)
(→Classification) |
(→Invariants) |
||
Line 20: | Line 20: | ||
== Invariants == | == Invariants == | ||
<wikitex>; | <wikitex>; | ||
− | Suppose that $M$ is diffeomorphic to $\ | + | Suppose that $M$ is diffeomorphic to $\sharp_b(S^3 \times S^3)$ then: |
− | * $\pi_3(M) \cong H_3(M) \cong \Zz^{ | + | * $\pi_3(M) \cong H_3(M) \cong \Zz^{2b}$, |
− | * the third Betti-number of $M$ is given by $b_3(M) = | + | * the third Betti-number of $M$ is given by $b_3(M) = 2b$, |
− | * the Euler characteristic of $M$ is given by $\chi(M) = 2 | + | * the Euler characteristic of $M$ is given by $\chi(M) = 2 - 2b$, |
− | * the [[Intersection forms|intersection form]] of $M$ is isomorphic to the sum of | + | * the [[Intersection forms|intersection form]] of $M$ is isomorphic to the sum of b-copies of $H_{-}(\Zz)$, the standard skew-symmetric hyperbolic form on $\Zz^2$. |
</wikitex> | </wikitex> | ||
Revision as of 11:31, 8 June 2010
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
Tex syntax error. 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
Tex syntax erroris diffeomorphic to a connected-sum
Tex syntax error
Tex syntax errorand in general is determined by the formula for the Euler characteristic of
Tex syntax error
- 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.
-
Tex syntax error
, the -fold connected sum of .
3 Invariants
Tex syntax erroris diffeomorphic to
Tex syntax errorthen:
- ,
- 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 b-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 denotes the natural numbers we obtain a bijection
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. That is, there is a bijection
Tex syntax error, and 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