Sandbox

From Manifold Atlas
Revision as of 17:06, 17 June 2010 by Diarmuid Crowley (Talk | contribs)
Jump to: navigation, search

The sandbox is the page where you can experiment with the wiki syntax. Feel free to write nonsense or clear the page whenever you want.

  • For the more general case where
    Tex syntax error
    , see 6-manifolds: 1-connected.

1 Introduction

Tex syntax error
Let
Tex syntax error
be the set of diffeomorphism classes of closed smooth simply-connected 2-connected 6-manifolds M. The classification
Tex syntax error
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 M is diffeomorphic to a connected-sum
Tex syntax error
where by definition
Tex syntax error
and in general r is determined by the formula for the Euler characteristic of M
Tex syntax error

1 Construction and examples

The following gives a complete list of 2-connected 6-manifolds up to diffeomorphism:

  • Tex syntax error
    , the standard 6-sphere.
  • Tex syntax error
    , the b-fold connected sum of S^3 \times S^3.

2 Invariants

Suppose that M is diffeomorphic to
Tex syntax error
then:
  • Tex syntax error
    ,
  • the third Betti-number of M is given by
    Tex syntax error
    ,
  • the Euler characteristic of M is given by
    Tex syntax error
    ,
  • the intersection form of M is isomorphic to the sum of b-copies of
    Tex syntax error
    , the standard skew-symmetric hyperbolic form on \Zz^2.

3 Classification

Recall that the following theorem was stated in other words in the introduction:

Theorem 7.1 [Smale1962a, Corolary 1.3]. The semi-group of 2-connected 6-manifolds is generated by S^3 \times S^3.

Hence if
Tex syntax error
denotes the natural numbers we obtain a bijection
Tex syntax error

4 Further discussion

4.1 Topological 2-connected 6-manifolds

Let
Tex syntax error
be the set of homeomorphism classes of topological 2-connected 6-manifolds.

Theorem 9.1. 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

Proof.

For any such manifold M we have
Tex syntax error
and so M is smoothable (see 6-manifolds: 1-connected). Any two homeomorphic manifolds have the same Euler Characteristic and so by Theorem 7.1 are diffeomorphic.
\square

4.2 Mapping class groups

...

2 References

Personal tools
Namespaces
Variants
Actions
Navigation
Interaction
Toolbox