Sandbox

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 $A H_1(M) \neq 0/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_kn5LBs$$Write here... <wikitex>; Poincare complexes are great: $$ H_i(X) \cong H^{n-i}(X)$$ </wikitex>\newcommand{\ZZZ}{\mathbb{Z}}A H_1(M) \neq 0$, see 6-manifolds: 1-connected.

• For the more general case where $B H_1(M) \neq 0/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_6GmX4D$$B H_1(M) \neq 0$, see 6-manifolds: 1-connected.

• For the more general case where $C H_1(M) \neq 0/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_MCNLXP$$C H_1(M) \neq 0$, see 6-manifolds: 1-connected.

• For the more general case where $E H_1(M) \neq 0/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_s0i4f2$$E H_1(M) \neq 0$, see 6-manifolds: 1-connected.

• For the more general case where $F H_1(M) \neq 0/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_quGwXe$$F H_1(M) \neq 0$, see 6-manifolds: 1-connected.

1 Introduction

$\ZZZ/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_Cauu4r$$\ZZZ$

$\displaystyle f = g/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_AUtWyF$ $$\displaystyle f = g$$

6.1 Let $\mathcal{M}_6(0)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_4UK8qT$$\mathcal{M}_6(0)$ be the set of diffeomorphism classes of closed smooth simply-connected 2-connected 6-manifolds $M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_aRW7G7$$M$.

The classification $\mathcal{M}_6(0)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_Ev3drm$$\mathcal{M}_6(0)$ was one of Smale's first applications of the h-cobordism theorem [Smale1962a, Corollary 1.3]. The classification, as for oriented surfaces is strikingly simple: every 2-connected 6-manifold $M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_qmmfBB$$M$ is diffeomorphic to a connected-sum

$\displaystyle M \cong \sharp_r(S^3 \times S^3)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_qnjw8Q$ $$\displaystyle M \cong \sharp_r(S^3 \times S^3)$$

where by definition $\sharp_0(S^3 \times S^3) = S^6/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_aLTx36$$\sharp_0(S^3 \times S^3) = S^6$ and in general $r/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_ClInmn$$r$ is determined by the formula for the Euler characteristic of $M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_CY8V1D$$M$

$\displaystyle \chi(M) = 2 - 2r./var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_INc04U$ $$\displaystyle \chi(M) = 2 - 2r.$$

• For the more general case where $H_2(M) \neq 0/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_WJSawc$$H_2(M) \neq 0$, see 6-manifolds: 1-connected.

Construction and examples $L^2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_KgYyqu$$L^2$

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_gQzCKM$$S^6$, the standard 6-sphere.
• $\sharp_b(S^3 \times S^3)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_AxQLt5$$\sharp_b(S^3 \times S^3)$, the $b/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_iQq7Bo$$b$-fold connected sum of $S^3 \times S^3/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_kBvU9H$$S^3 \times S^3$.

2 Invariants

Suppose that $M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_SQuM71$$M$ is diffeomorphic to $\sharp_b(S^3 \times S^3)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_4PjYtm$$\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_STfqkH$$\pi_3(M) \cong H_3(M) \cong \Zz^{2b}$,
• the third Betti-number of $M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_QWzwA2$$M$ is given by $b_3(M) = 2b/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_qbUEfo$$b_3(M) = 2b$,
• the Euler characteristic of $M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_2JLpjK$$M$ is given by $\chi(M) = 2 - 2b/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_y0yiL6$$\chi(M) = 2 - 2b$,
• the intersection form of $M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_seJXBt$$M$ is isomorphic to the sum of b-copies of $H_{-}(\Zz)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_shTiRQ$$H_{-}(\Zz)$, the standard skew-symmetric hyperbolic form on $\Zz^2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_Qe0Tve$$\Zz^2$.

Classification

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

Theorem 5.1 [Smale1962a, Corolary 1.3]. The semi-group of 2-connected 6-manifolds is generated by $S^3 \times S^3$$S^3 \times S^3$.

Hence if $\Nn$$\Nn$ denotes the natural numbers we obtain a bijection

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

3 Further discussion

3.1 Topological 2-connected 6-manifolds

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

Theorem 6.1. Every topological 2-connected 6-manifold admits a smooth structure which is unique up to diffoemorphism. That is, there is a bijection

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

Proof. For any such manifold $M$$M$ we have $H^4(M; \Zz/2) \cong 0$$H^4(M; \Zz/2) \cong 0$ and so $M$$M$ is smoothable (see 6-manifolds: 1-connected). Any two homeomorphic manifolds have the same Euler Characteristic and so by Theorem 5.1 are diffeomorphic.

$\square$$\square$

Mapping class groups

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4 References

• [Smale1962a] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR0153022 (27 #2991) Zbl 0109.41103