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- For the more general case where
, see 6-manifolds: 1-connected.
- For the more general case where
, see 6-manifolds: 1-connected.
- For the more general case where
, see 6-manifolds: 1-connected.
- For the more general case where
, see 6-manifolds: 1-connected.
1 Introduction
Let be the set of diffeomorphism classes of closed smooth simply-connected 2-connected 6-manifolds
.
The classification 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
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_mj1Wbd](/images/math/e/3/7/e3702a339d68030757da3a7846b3f1dc.png)
where by definition and in general
is determined by the formula for the Euler characteristic of
![\displaystyle \chi(M) = 2 - 2r./var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_UsQd6K](/images/math/9/c/9/9c9cd92233e15d5523da63c3fd1322d0.png)
- For the more general case where
, see 6-manifolds: 1-connected.
1 Construction and examples
The following gives a complete list of 2-connected 6-manifolds up to diffeomorphism:
-
, the standard 6-sphere.
-
, the
-fold connected sum of
.
2 Invariants
Suppose that is diffeomorphic to
then:
-
,
- the third Betti-number of
is given by
,
- the Euler characteristic of
is given by
,
- the intersection form of
is isomorphic to the sum of b-copies of
, the standard skew-symmetric hyperbolic form on
.
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 .
Hence if 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_maaFfw](/images/math/4/f/8/4f87caab28b3b9b7fd16a5f88c9bb677.png)
4 Further discussion
4.1 Topological 2-connected 6-manifolds
Let 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_guU9u4](/images/math/5/4/7/54760c11b8f0267445e9ec57732484ef.png)
Proof.
For any such manifold we have
and so
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](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
4.2 Mapping class groups
...
2 References
- [Smale1962a] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR0153022 (27 #2991) Zbl 0109.41103