Sandbox
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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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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 $\Zz^2$. | ||
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== Classification == | == Classification == | ||
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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 | ||
$$ \mathcal{M}_6(0)\equiv \Nn,~~~[M] \mapsto \frac{1}{2}b_3(M).$$ | $$ \mathcal{M}_6(0)\equiv \Nn,~~~[M] \mapsto \frac{1}{2}b_3(M).$$ | ||
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== Further discussion == | == Further discussion == | ||
=== Topological 2-connected 6-manifolds === | === Topological 2-connected 6-manifolds === | ||
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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}} | ||
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=== Mapping class groups === | === Mapping class groups === | ||
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... | ... | ||
<wikitex> | <wikitex> |
Revision as of 10:07, 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.
1 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
.
2 Invariants
Suppose thatTex syntax erroris diffeomorphic to
Tex syntax errorthen:
-
Tex syntax error
, - the third Betti-number of
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is given byTex syntax error
, - the Euler characteristic of
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is given byTex syntax error
, - the intersection form of
Tex syntax error
is isomorphic to the sum of b-copies ofTex syntax error
, the standard skew-symmetric hyperbolic form onTex syntax error
.
3 Classification
Recall that the following theorem was stated in other words in the introduction:
Theorem 11.1 [Smale1962a, Corolary 1.3].
The semi-group of 2-connected 6-manifolds is generated byTex syntax error.
Tex syntax errordenotes the natural numbers we obtain a bijection
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4 Further discussion
4.1 Topological 2-connected 6-manifolds
LetTex syntax errorbe the set of homeomorphism classes of topological 2-connected 6-manifolds.
Theorem 14.1. Every topological 2-connected 6-manifold admits a smooth structure which is unique up to diffoemorphism. That is, there is a bijection
Tex syntax error
Proof.
For any such manifoldTex syntax errorwe have
Tex syntax errorand so
Tex syntax erroris smoothable (see 6-manifolds: 1-connected). Any two homeomorphic manifolds have the same Euler Characteristic and so by Theorem 11.1 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