Sandbox

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• For the more general case where $A H_1(M) \neq 0/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_WOA7MO$$<wikitex refresh>; * For the more general case where $A H_1(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]]. </wikitex> <wikitex refresh include="MediaWiki:MathFontCM">; * For the more general case where $B H_1(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]]. </wikitex> <wikitex refresh include="MediaWiki:MathFontHV">; * For the more general case where $C H_1(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]]. </wikitex> <wikitex refresh include="MediaWiki:MathFont">; * For the more general case where $E H_1(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]]. </wikitex> <wikitex refresh include="MediaWiki:MathFontLM">; * For the more general case where $F H_1(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]]. </wikitex> == Introduction == <wikitex refresh include = "TexInclude:Sandbox; MediaWiki:MathFontCM">; $\ZZZ$ $$ f = g$$ \ref{test} Let $\mathcal{M}_6(0)$ be the set of diffeomorphism classes of [[wikipedia:Closed_manifold|closed]] [[wikipedia:Differentiable_manifold|smooth]] [[wikipedia:Simply-connected|simply-connected]] 2-connected 6-manifolds $M$. The classification $\mathcal{M}_6(0)$ was one of Smale's first applications of the [[Wikipedia:h-cobordism_theorem|h-cobordism]] theorem {{cite|Smale1962a|Corollary 1.3}}. The classification, as for [[Surface|oriented surfaces]] is strikingly simple: every 2-connected 6-manifold $M$ is diffeomorphic to a [[Wikipedia:Connected-sum|connected-sum]] $$ M \cong \sharp_r(S^3 \times S^3)$$ where by definition $\sharp_0(S^3 \times S^3) = S^6$ and in general $r$ is determined by the formula for the [[Wikipedia:Euler characteristic|Euler characteristic]] of $M$ $$ \chi(M) = 2 - 2r.$$ * For the more general case where $H_2(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]]. </wikitex> <wikitex refresh include="MediaWiki:MathFontCM">; == Construction and examples $L^2$ == The following gives a complete list of 2-connected 6-manifolds up to diffeomorphism: * $S^6$, the standard 6-sphere. * $\sharp_b(S^3 \times S^3)$, the $b$-fold connected sum of $S^3 \times S^3$. </wikitex> == Invariants == <wikitex refresh include="MediaWiki:MathFontHV">; Suppose that $M$ is diffeomorphic to $\sharp_b(S^3 \times S^3)$ then: * $\pi_3(M) \cong H_3(M) \cong \Zz^{2b}$, * the third Betti-number of $M$ is given by $b_3(M) = 2b$, * the Euler characteristic of $M$ is given by $\chi(M) = 2 - 2b$, * the [[Intersection forms|intersection form]] of $M$ is isomorphic to the sum of b-copies of $H_{-}(\Zz)$, the standard skew-symmetric hyperbolic form on $\Zz^2$. </wikitex> <wikitex> == Classification == Recall that the following theorem was stated in other words in the introduction: {{beginthm|Theorem|{{cite|Smale1962a|Corolary 1.3}}}} \label{thm:classification} The semi-group of 2-connected 6-manifolds is generated by $S^3 \times S^3$. {{endthm}} Hence if $\Nn$ denotes the natural numbers we obtain a bijection $$ \mathcal{M}_6(0)\equiv \Nn,[M] \mapsto \frac{1}{2}b_3(M).$$ </wikitex> == Further discussion == === Topological 2-connected 6-manifolds === <wikitex>; Let $\mathcal{M}^{\Top}_6(e)$ be the set of homeomorphism classes of topological 2-connected 6-manifolds. {{beginthm|Theorem}}\label{test} Every topological 2-connected 6-manifold admits a smooth structure which is unique up to diffoemorphism. That is, there is a bijection $$ \mathcal{M}_6(e) \rightarrow \mathcal{M}^{\Top}_6(e).$$ {{endthm}} {{beginproof}} For any such manifold $M$ we have $H^4(M; \Zz/2) \cong 0$ and so $M$ is smoothable (see [[6-manifolds: 1-connected#Smoothing theory|6-manifolds: 1-connected]]). Any two homeomorphic manifolds have the same Euler Characteristic and so by Theorem \ref{thm:classification} are diffeomorphic. {{endproof}} <wikitex>; === Mapping class groups === ... </wikitex> == References == {{#RefList:}}\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_UgeDZJ$$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_6tyhCF$$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_W4LFFB$$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_g5uKhy$$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_GAlWxv$$\ZZZ$

$\displaystyle f = g/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_0PCPct$ $$\displaystyle f = g$$

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

The classification $\mathcal{M}_6(0)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_q7r8tp$$\mathcal{M}_6(0)$ was one of Smale's first applications of the h-cobordism theorem [Smale1962a, Corollary 1.3]. The classification, as for oriented surfaces is strikingly simple: every 2-connected 6-manifold $M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_i0uhbp$$M$ is diffeomorphic to a connected-sum

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

where by definition $\sharp_0(S^3 \times S^3) = S^6/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_oxuXHo$$\sharp_0(S^3 \times S^3) = S^6$ and in general $r/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_AlZxEp$$r$ is determined by the formula for the Euler characteristic of $M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_gkoC7q$$M$

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

• For the more general case where $H_2(M) \neq 0/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_85GGfv$$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_QoeMUx$$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_IFYTWA$$S^6$, the standard 6-sphere.
• $\sharp_b(S^3 \times S^3)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_a0B1lE$$\sharp_b(S^3 \times S^3)$, the $b/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_mRHE9H$$b$-fold connected sum of $S^3 \times S^3/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_uu7kkM$$S^3 \times S^3$.

2 Invariants

Suppose that $M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_cpXqTQ$$M$ is diffeomorphic to $\sharp_b(S^3 \times S^3)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_Cwk8PV$$\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_KT2La1$$\pi_3(M) \cong H_3(M) \cong \Zz^{2b}$,
• the third Betti-number of $M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_QvMNT6$$M$ is given by $b_3(M) = 2b/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_c7YR0c$$b_3(M) = 2b$,
• the Euler characteristic of $M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_gqFQwj$$M$ is given by $\chi(M) = 2 - 2b/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_yoSOpq$$\chi(M) = 2 - 2b$,
• the intersection form of $M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_65Y0Fx$$M$ is isomorphic to the sum of b-copies of $H_{-}(\Zz)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_C4wRiF$$H_{-}(\Zz)$, the standard skew-symmetric hyperbolic form on $\Zz^2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_8SZAoN$$\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)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_SSZ7UV$$\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

$\displaystyle \mathcal{M}_6(e) \rightarrow \mathcal{M}^{\Top}_6(e)./var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_YGPJP4$ $$\displaystyle \mathcal{M}_6(e) \rightarrow \mathcal{M}^{\Top}_6(e).$$

Proof. For any such manifold $M$$M$ we have $H^4(M; \Zz/2) \cong 0/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_QhmB9d$$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

...

4 References

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