6-manifolds: 2-connected

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== Invariants ==
== Invariants ==
<wikitex>;
<wikitex>;
Suppose that $M$ is diffeomorphic to $\sharp_r(S^3 \times S^3)$ then:
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Suppose that $M$ is diffeomorphic to $\sharp_b(S^3 \times S^3)$ then:
* $\pi_3(M) \cong H_3(M) \cong \Zz^{2r}$,
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* $\pi_3(M) \cong H_3(M) \cong \Zz^{2b}$,
* the third Betti-number of $M$ is given by $b_3(M) = 2r$,
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* the third Betti-number of $M$ is given by $b_3(M) = 2b$,
* the Euler characteristic of $M$ is given by $\chi(M) = 2 = 2r$,
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* 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 r-copies of $H_{-}(\Zz)$, the standard skew-symmetric hyperbolic form on $\Zz^2$.
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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 $\Zz^2$.
</wikitex>
</wikitex>

Revision as of 11:31, 8 June 2010

This page has not been refereed. The information given here might be incomplete or provisional.

Contents

1 Introduction

Let \mathcal{M}_6(0) be the set of diffeomorphism classes of closed smooth simply-connected 2-connected 6-manifolds
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. The classification \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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is diffeomorphic to a connected-sum
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where by definition
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and in general r is determined by the formula for the Euler characteristic of
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\displaystyle  \chi(M) = 2 - 2r.

2 Construction and examples

The following gives a complete list of 2-connected 6-manifolds up to diffeomorphism:

  • S^6, the standard 6-sphere.
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    , the b-fold connected sum of S^3 \times S^3.

3 Invariants

Suppose that
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is diffeomorphic to
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then:
  • \pi_3(M) \cong H_3(M) \cong \Zz^{2b},
  • the third Betti-number of
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    is given by b_3(M) = 2b,
  • the Euler characteristic of
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    is given by \chi(M) = 2 - 2b,
  • the intersection form of
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    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

\displaystyle  \mathcal{M}_6(0)\equiv \Nn,~~~[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

\displaystyle  \mathcal{M}_6(e) \rightarrow \mathcal{M}^{\Top}_6(e).
The above follows since for any such
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, H^4(M; \Zz/2) \cong 0 and so 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

...


References

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