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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.

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Introduction

Let \mathcal{M}_6(0)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_R3GsOr be the set of diffeomorphism classes of closed smooth simply-connected 2-connected 6-manifolds M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_TZ5rki.

The classification \mathcal{M}_6(0)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_pObSg9 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/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_TcLMD0 is diffeomorphic to a connected-sum

\displaystyle  M \cong \sharp_r(S^3 \times S^3)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_pViipS

where by definition \sharp_0(S^3 \times S^3) = S^6/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_1KNoEK and in general r/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_RVkujD is determined by the formula for the Euler characteristic of M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_f2fUnw

\displaystyle  \chi(M) = 2 - 2r./var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_LmOqRp

1 Construction and examples

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

  • S^6/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_JcZQ2d, the standard 6-sphere.
  • \sharp_b(S^3 \times S^3)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_Fvi6P8, the b/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_ZNfz33-fold connected sum of S^3 \times S^3/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_NB8pHZ.

2 Invariants

Suppose that M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_FDGVKV is diffeomorphic to \sharp_b(S^3 \times S^3)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_BCWidS then:

  • \pi_3(M) \cong H_3(M) \cong \Zz^{2b}/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_LZFr5O,
  • the third Betti-number of M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_5m2BqM is given by b_3(M) = 2b/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_PXxNbK,
  • the Euler characteristic of M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_pIZ0lI is given by \chi(M) = 2 - 2b/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_D7CmUG,
  • the intersection form of M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_LKtxRF is isomorphic to the sum of b-copies of H_{-}(\Zz)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_FyKYcF, the standard skew-symmetric hyperbolic form on \Zz^2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_xVbr1E.

3 Classification

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

Theorem 11.1 [Smale1962a, Corolary 1.3]. The semi-group of 2-connected 6-manifolds is generated by S^3 \times S^3/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_BZdkgF.

Hence if \Nn/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_ni5OVF denotes the natural numbers we obtain a bijection

\displaystyle  \mathcal{M}_6(0)\equiv \Nn,~~~[M] \mapsto \frac{1}{2}b_3(M)./var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_3IjO1G

4 Further discussion

4.1 Topological 2-connected 6-manifolds

Let \mathcal{M}^{\Top}_6(e)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_bv8vyI be the set of homeomorphism classes of topological 2-connected 6-manifolds.

Theorem 14.1. 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_VlAJuK

Proof. For any such manifold M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_xc3pRM we have H^4(M; \Zz/2) \cong 0/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_7AsuDP and so M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_V6ExPS is smoothable (see 6-manifolds: 1-connected). Any two homeomorphic manifolds have the same Euler Characteristic and so by Theorem 11.1 are diffeomorphic.

\square

4.2 Mapping class groups

...

References

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