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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
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    , see 6-manifolds: 1-connected.

1 Introduction

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

2 Invariants

Suppose that
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is diffeomorphic to
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then:
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    ,
  • the third Betti-number of
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    is given by
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    ,
  • the Euler characteristic of
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    is given by
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    ,
  • the intersection form of
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    is isomorphic to the sum of b-copies of
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    , the standard skew-symmetric hyperbolic form on
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    .

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
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.
Hence if
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denotes the natural numbers we obtain a bijection
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4 Further discussion

4.1 Topological 2-connected 6-manifolds

Let
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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

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Proof.

For any such manifold
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we have
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and so
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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

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