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	$$ \chi(M) = 2 - 2r.$$		$$ \chi(M) = 2 - 2r.$$
	* For the more general case where $H_2(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]].		* For the more general case where $H_2(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]].
−	</wikitex>
	== Construction and examples ==		== Construction and examples ==
−	<wikitex>;
	The following gives a complete list of 2-connected 6-manifolds up to diffeomorphism:		The following gives a complete list of 2-connected 6-manifolds up to diffeomorphism:
	* $S^6$, the standard 6-sphere.		* $S^6$, the standard 6-sphere.
	* $\sharp_b(S^3 \times S^3)$, the $b$-fold connected sum of $S^3 \times S^3$.		* $\sharp_b(S^3 \times S^3)$, the $b$-fold connected sum of $S^3 \times S^3$.
−	</wikitex>
	== Invariants ==		== Invariants ==
−	<wikitex>;
	Suppose that $M$ is diffeomorphic to $\sharp_b(S^3 \times S^3)$ then:		Suppose that $M$ is diffeomorphic to $\sharp_b(S^3 \times S^3)$ then:
	* $\pi_3(M) \cong H_3(M) \cong \Zz^{2b}$,		* $\pi_3(M) \cong H_3(M) \cong \Zz^{2b}$,
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	* the Euler characteristic of $M$ is given by $\chi(M) = 2 - 2b$,		* 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 $\Zz^2$.		* 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 ==		== Classification ==
−	<wikitex>;
	Recall that the following theorem was stated in other words in the introduction:		Recall that the following theorem was stated in other words in the introduction:
	{{beginthm|Theorem|{{cite|Smale1962a|Corolary 1.3}}}} \label{thm:classification}		{{beginthm|Theorem|{{cite|Smale1962a|Corolary 1.3}}}} \label{thm:classification}
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	Hence if $\Nn$ denotes the natural numbers we obtain a bijection		Hence if $\Nn$ denotes the natural numbers we obtain a bijection
	$$ \mathcal{M}_6(0)\equiv \Nn,~~~[M] \mapsto \frac{1}{2}b_3(M).$$		$$ \mathcal{M}_6(0)\equiv \Nn,~~~[M] \mapsto \frac{1}{2}b_3(M).$$
−	</wikitex>
	== Further discussion ==		== Further discussion ==
	=== Topological 2-connected 6-manifolds ===		=== Topological 2-connected 6-manifolds ===
−	<wikitex>;
	Let $\mathcal{M}^{\Top}_6(e)$ be the set of homeomorphism classes of topological 2-connected 6-manifolds.		Let $\mathcal{M}^{\Top}_6(e)$ be the set of homeomorphism classes of topological 2-connected 6-manifolds.
	{{beginthm|Theorem}}		{{beginthm|Theorem}}
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	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.		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}}		{{endproof}}
−	</wikitex>
	=== Mapping class groups ===		=== Mapping class groups ===
−	<wikitex>;
	...		...
	<wikitex>		<wikitex>

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Revision as of 10:07, 11 June 2010

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Introduction

Let

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$<wikitex refresh include="MediaWiki:MathFont">; Write here... == 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 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> == 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 $\mathcal{M}^{\Top}_6(e)$ 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. 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>; ... <wikitex> == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Highly-connected manifolds]]<wikitex refresh include="MediaWiki:MathFont">; Write here... == 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 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> == 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 $\mathcal{M}^{\Top}_6(e)$ 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. 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>; ... <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

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$M$. 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 classification, as for oriented surfaces is strikingly simple: every 2-connected 6-manifold

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$M$ 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

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$r$ is determined by the formula for the Euler characteristic of

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$M$

• For the more general case where

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

1 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

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  $b$-fold connected sum of

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  $S^3 \times S^3$.

2 Invariants

Suppose that

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$M$ 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

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  $M$ is given by

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

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  $M$ is given by

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  $\chi(M) = 2 - 2b$,
• the intersection form of

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  $M$ is isomorphic to the sum of b-copies of

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

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  $\Zz^2$.

3 Classification

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

Hence if

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

4 Further discussion

4.1 Topological 2-connected 6-manifolds

Let

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

4.2 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