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  • For the more general case where A H_1(M) \neq 0, see 6-manifolds: 1-connected.

1 Introduction

\ZZZ/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_JkZsjW Let \mathcal{M}_6(0) be the set of diffeomorphism classes of closed smooth simply-connected 2-connected 6-manifolds M.

The classification \mathcal{M}_6(0) 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

\displaystyle  M \cong \sharp_r(S^3 \times S^3)

where by definition \sharp_0(S^3 \times S^3) = S^6 and in general r is determined by the formula for the Euler characteristic of M

\displaystyle  \chi(M) = 2 - 2r.

1 Construction and examples

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

  • S^6, the standard 6-sphere.
  • \sharp_b(S^3 \times S^3)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_NUDTIL, the b-fold connected sum of S^3 \times S^3.

2 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}/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_DaXmAB,
  • the third Betti-number of M is given by b_3(M) = 2b/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_jVowSr,
  • the Euler characteristic of M is given by \chi(M) = 2 - 2b/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_TINdBi,
  • 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.

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 \Nn denotes the natural numbers we obtain a bijection

\displaystyle  \mathcal{M}_6(0)\equiv \Nn,~~~[M] \mapsto \frac{1}{2}b_3(M).

4 Further discussion

4.1 Topological 2-connected 6-manifolds

Let \mathcal{M}^{\Top}_6(e) be the set of homeomorphism classes of topological 2-connected 6-manifolds.

Theorem 9.1. Every topological 2-connected 6-manifold admits a smooth structure which is unique up to diffoemorphism. That is, there is a bijection

\displaystyle  \mathcal{M}_6(e) \rightarrow \mathcal{M}^{\Top}_6(e)./var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_begKL9

Proof. For any such manifold M we have H^4(M; \Zz/2) \cong 0/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_1EXIn1 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

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