6-manifolds: 2-connected

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	=== Topological 2-connected 6-manifolds ===		=== Topological 2-connected 6-manifolds ===
	<wikitex>;		<wikitex>;
−	Let $\mathcal{M}^{\Top}_6$ be the set of homeomorphism classes of topological 2-connected 6-manifolds.	+	Let $\mathcal{M}^{\Top}_6(0)$ be the set of homeomorphism classes of topological 2-connected 6-manifolds.
	{{beginthm|Theorem}}		{{beginthm|Theorem}}
	Every topological 2-connected 6-manifold admits a smooth structure which is unique up to diffoemorphism. In particular, there is a bijection		Every topological 2-connected 6-manifold admits a smooth structure which is unique up to diffoemorphism. In particular, there is a bijection
−	$$ \mathcal{M}_6(0) \rightarrow \mathcal{M}^{\Top}_6.$$	+	$$ \mathcal{M}_6(0) \rightarrow \mathcal{M}^{\Top}_6(0).$$
	{{endthm}}		{{endthm}}

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Revision as of 12:00, 27 November 2010

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Contents 1 Introduction 2 Construction and examples 3 Invariants 4 Classification 5 Further discussion 5.1 Topological 2-connected 6-manifolds 5.2 Mapping class groups

1 Introduction

Let

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${{Stub}} == Introduction == <wikitex>; 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 notation is used to be consistent with [[6-manifolds: 1-connected]]). 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 is a precise 6-dimensional analogue of the classification of [[Surface|orientable surfaces]]: 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> == Construction and examples == <wikitex>; 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>; 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> == Classification == <wikitex>; 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 $\mathcal{M}^{\Top}_6$ be the set of homeomorphism classes of topological 2-connected 6-manifolds. {{beginthm|Theorem}} Every topological 2-connected 6-manifold admits a smooth structure which is unique up to diffoemorphism. In particular, there is a bijection $$ \mathcal{M}_6(0) \rightarrow \mathcal{M}^{\Top}_6.$$ {{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>; ... <wikitex> == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Highly-connected manifolds]]\mathcal{M}_6(0)$ be the set of diffeomorphism classes of closed smooth simply-connected 2-connected 6-manifolds $M$ (the notation is used to be consistent with 6-manifolds: 1-connected). The classification

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$\mathcal{M}_6(0)$ was one of Smale's first applications of the h-cobordism theorem [Smale1962a, Corollary 1.3]. The is a precise 6-dimensional analogue of the classification of orientable surfaces: every 2-connected 6-manifold $M$ is diffeomorphic to a connected-sum where by definition

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

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$H_2(M) \neq 0$, see 6-manifolds: 1-connected.

2 Construction and examples

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

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  $S^6$, the standard 6-sphere.

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  $\sharp_b(S^3 \times S^3)$, the $b$-fold connected sum of

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  $S^3 \times S^3$.

3 Invariants

Suppose that $M$ is diffeomorphic to

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$\sharp_b(S^3 \times S^3)$ then:

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  $\pi_3(M) \cong H_3(M) \cong \Zz^{2b}$,
• the third Betti-number of $M$ is given by

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  $b_3(M) = 2b$,
• the Euler characteristic of $M$ is given by

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  $\chi(M) = 2 - 2b$,
• the intersection form of $M$ is isomorphic to the sum of b-copies of

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  $H_{-}(\Zz)$, the standard skew-symmetric hyperbolic form on $\Zz^2$.

4 Classification

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

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

5 Further discussion

5.1 Topological 2-connected 6-manifolds

Let $\mathcal{M}^{\Top}_6(0)$ be the set of homeomorphism classes of topological 2-connected 6-manifolds.

5.2 Mapping class groups

...

References

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