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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]]. | ||
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== Construction and examples == | == Construction and examples == | ||
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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 $S^3 \times S^3$. | ||
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== Invariants == | == Invariants == | ||
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{{endthm}} | {{endthm}} | ||
Hence if $\Nn$ denotes the natural numbers we obtain a bijection | Hence if $\Nn$ denotes the natural numbers we obtain a bijection | ||
− | $$ \mathcal{M}_6(0)\equiv \Nn, | + | $$ \mathcal{M}_6(0)\equiv \Nn,[[User:Diarmuid Crowley|Diarmuid Crowley]][M] \mapsto \frac{1}{2}b_3(M).$$ |
== Further discussion == | == Further discussion == |
Revision as of 18:07, 17 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. |
- For the more general case where
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, see 6-manifolds: 1-connected.
- For the more general case where
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, see 6-manifolds: 1-connected.
- For the more general case where
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, see 6-manifolds: 1-connected.
- For the more general case where
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, see 6-manifolds: 1-connected.
- For the more general case where
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, see 6-manifolds: 1-connected.
Contents |
1 Introduction
Tex syntax errorLet
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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 is diffeomorphic to a connected-sum
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Tex syntax errorand in general is determined by the formula for the Euler characteristic of
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- For the more general case where
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, 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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, the standard 6-sphere. -
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, theTex syntax error
-fold connected sum ofTex syntax error
.
3 Invariants
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 form of $M$ is isomorphic to the sum of b-copies of $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:
Theorem 4.1 [Smale1962a, Corolary 1.3]. The semi-group of 2-connected 6-manifolds is generated by $S^3 \times S^3$.
Hence if $\Nn$ denotes the natural numbers we obtain a bijection $$ \mathcal{M}_6(0)\equiv \Nn,Diarmuid Crowley[M] \mapsto \frac{1}{2}b_3(M).$$
5 Further discussion
5.1 Topological 2-connected 6-manifolds
Let $\mathcal{M}^{\Top}_6(e)$ be the set of homeomorphism classes of topological 2-connected 6-manifolds.
Theorem 5.1. Every topological 2-connected 6-manifold admits a smooth structure which is unique up to diffoemorphism. That is, there is a bijection $$ \mathcal{M}_6(e) \rightarrow \mathcal{M}^{\Top}_6(e).$$
Proof. For any such manifold $M$ we have $H^4(M; \Zz/2) \cong 0$ and so $M$ is smoothable (see 6-manifolds: 1-connected). Any two homeomorphic manifolds have the same Euler Characteristic and so by Theorem 4.1 are diffeomorphic.
5.2 Mapping class groups
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6 References
- [Smale1962a] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR0153022 (27 #2991) Zbl 0109.41103