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− | $ | + | == 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]] | |
− | {{beginthm| | + | $$ M \cong \sharp_r(S^3 \times S^3)$$ |
+ | 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}} | {{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> | |
− | {{beginthm| | + | |
+ | == 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}} | {{endthm}} | ||
+ | |||
{{beginproof}} | {{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}} | {{endproof}} | ||
− | + | </wikitex> | |
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | === Mapping class groups === | |
+ | <wikitex>; | ||
+ | ... | ||
+ | <wikitex> | ||
+ | == References == | ||
+ | {{#RefList:}} | ||
− | + | [[Category:Manifolds]] | |
− | + | [[Category:Highly-connected manifolds]] | |
− | + | ||
− | + | ||
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Revision as of 10:06, 11 June 2010
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. |
Write here...
Introduction
Tex syntax errorbe the set of diffeomorphism classes of closed smooth simply-connected 2-connected 6-manifolds
Tex syntax error. The classification
Tex syntax errorwas 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
Tex syntax erroris diffeomorphic to a connected-sum
Tex syntax error
Tex syntax errorand in general
Tex syntax erroris determined by the formula for the Euler characteristic of
Tex syntax error
Tex syntax error
- For the more general case where
Tex syntax error
, see 6-manifolds: 1-connected.
Contents |
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
Tex syntax erroris diffeomorphic to then:
- ,
- the third Betti-number of
Tex syntax error
is given by , - the Euler characteristic of
Tex syntax error
is given by , - the intersection form of
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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 5.1 [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
Tex syntax error
4 Further discussion
4.1 Topological 2-connected 6-manifolds
Let 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
Proof.
For any such manifoldTex syntax errorwe have and so
Tex syntax erroris smoothable (see 6-manifolds: 1-connected). Any two homeomorphic manifolds have the same Euler Characteristic and so by Theorem 5.1 are diffeomorphic.
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