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− | * For the more general case where $A H_1(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]].
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− | * For the more general case where $B H_1(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]].
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− | * For the more general case where $C H_1(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]].
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− | * For the more general case where $E H_1(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]].
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− | * For the more general case where $F H_1(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]].
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− | </wikitex>
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− | == Introduction ==
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− | <wikitex refresh include = "TexInclude:Sandbox; MediaWiki:MathFontCM">;
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− | $\ZZZ$
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− | $$ f = g$$
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− | \ref{test}
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− | 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 $M$.
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− | 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]]
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− | $$ M \cong \sharp_r(S^3 \times S^3)$$
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− | 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 $M$
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− | $$ \chi(M) = 2 - 2r.$$
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− | * For the more general case where $H_2(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]].
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− | </wikitex>
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− | == Construction and examples $L^2$ ==
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− | 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 $S^3 \times S^3$.
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− | </wikitex>
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− | == Invariants ==
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− | Suppose that $M$ is diffeomorphic to $\sharp_b(S^3 \times S^3)$ then:
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− | * $\pi_3(M) \cong H_3(M) \cong \Zz^{2b}$,
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− | * the third Betti-number of $M$ is given by $b_3(M) = 2b$,
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− | * the Euler characteristic of $M$ is given by $\chi(M) = 2 - 2b$,
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− | * 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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− | </wikitex>
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− | <wikitex>
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− | == Classification ==
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− | Recall that the following theorem was stated in other words in the introduction:
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− | {{beginthm|Theorem|{{cite|Smale1962a|Corolary 1.3}}}} \label{thm:classification}
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− | The semi-group of 2-connected 6-manifolds is generated by $S^3 \times S^3$.
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− | {{endthm}}
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− | Hence if $\Nn$ denotes the natural numbers we obtain a bijection
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− | $$ \mathcal{M}_6(0)\equiv \Nn,[M] \mapsto \frac{1}{2}b_3(M).$$
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− | </wikitex>
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− |
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− | == Further discussion ==
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− | === Topological 2-connected 6-manifolds ===
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| <wikitex>; | | <wikitex>; |
− | Let $\mathcal{M}^{\Top}_6(e)$ be the set of homeomorphism classes of topological 2-connected 6-manifolds.
| + | Poincare complexes are great: |
− | {{beginthm|Theorem}}\label{test}
| + | $$ H_i(X) \cong H^{n-i}(X)$$ |
− | Every topological 2-connected 6-manifold admits a smooth structure which is unique up to diffoemorphism. That is, there is a bijection
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− | $$ \mathcal{M}_6(e) \rightarrow \mathcal{M}^{\Top}_6(e).$$ | + | |
− | {{endthm}}
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− | | + | |
− | {{beginproof}}
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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.
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− | {{endproof}}
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− | <wikitex>;
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− | | + | |
− | === Mapping class groups ===
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− | ...
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| </wikitex> | | </wikitex> |
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− | == References ==
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− | {{#RefList:}}
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