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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.

Contents

1 Introduction

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

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

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    , the standard 6-sphere.
  • Tex syntax error
    , the
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    -fold connected sum of
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    .

3 Invariants

Suppose that $M$ is diffeomorphic to $\sharp_b(S^3 \times S^3)$ then:

  • $\pi_3(M) \cong H_3(M) \cong \Zz^{2b}$,
  • the third Betti-number of $M$ is given by $b_3(M) = 2b$,
  • the Euler characteristic of $M$ is given by $\chi(M) = 2 - 2b$,
  • the intersection form of $M$ is isomorphic to the sum of b-copies of $H_{-}(\Zz)$, the standard skew-symmetric hyperbolic form on $\Zz^2$.

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 $S^3 \times S^3$.

Hence if $\Nn$ denotes the natural numbers we obtain a bijection $$ \mathcal{M}_6(0)\equiv \Nn,Diarmuid Crowley[M] \mapsto \frac{1}{2}b_3(M).$$

5 Further discussion

5.1 Topological 2-connected 6-manifolds

Let $\mathcal{M}^{\Top}_6(e)$ 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 $$ \mathcal{M}_6(e) \rightarrow \mathcal{M}^{\Top}_6(e).$$

Proof. For any such manifold $M$ we have $H^4(M; \Zz/2) \cong 0$ and so $M$ is smoothable (see 6-manifolds: 1-connected). Any two homeomorphic manifolds have the same Euler Characteristic and so by Theorem 4.1 are diffeomorphic.

\square

5.2 Mapping class groups

... </wikitex>

6 References

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