6-manifolds: 2-connected
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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 is a precise 6-dimensional analogue of the classification of [[Surface|orientable surfaces]]: every 2-connected 6-manifold $M$ is diffeomorphic to a [[Wikipedia:Connected-sum|connected-sum]] | 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 is a precise 6-dimensional analogue of the classification of [[Surface|orientable surfaces]]: every 2-connected 6-manifold $M$ is diffeomorphic to a [[Wikipedia:Connected-sum|connected-sum]] | ||
− | $$ M \cong \ | + | $$ M \cong \#_r(S^3 \times S^3)$$ |
− | where by definition $\ | + | where by definition $\#_0(S^3 \times S^3) = S^6$ and in general $r$ is determined by the formula for the [[Wikipedia:Euler characteristic|Euler characteristic]] of $M$ |
$$ \chi(M) = 2 - 2r.$$ | $$ \chi(M) = 2 - 2r.$$ | ||
For the more general case where $H_2(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]]. | For the more general case where $H_2(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]]. | ||
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The following gives a complete list of 2-connected 6-manifolds up to diffeomorphism: | The following gives a complete list of 2-connected 6-manifolds up to diffeomorphism: | ||
* $S^6$, the standard 6-sphere. | * $S^6$, the standard 6-sphere. | ||
− | * $\ | + | * $\#_b(S^3 \times S^3)$, the $b$-fold connected sum of $S^3 \times S^3$. |
</wikitex> | </wikitex> | ||
== Invariants == | == Invariants == | ||
<wikitex>; | <wikitex>; | ||
− | Suppose that $M$ is diffeomorphic to $\ | + | Suppose that $M$ is diffeomorphic to $\#_b(S^3 \times S^3)$ then: |
* $\pi_3(M) \cong H_3(M) \cong \Zz^{2b}$, | * $\pi_3(M) \cong H_3(M) \cong \Zz^{2b}$, | ||
* the third Betti-number of $M$ is given by $b_3(M) = 2b$, | * the third Betti-number of $M$ is given by $b_3(M) = 2b$, | ||
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{{beginthm|Theorem}} | {{beginthm|Theorem}} | ||
Every topological 2-connected 6-manifold admits a smooth structure which is unique up to diffoemorphism. In particular, there is a bijection | Every topological 2-connected 6-manifold admits a smooth structure which is unique up to diffoemorphism. In particular, there is a bijection | ||
− | $$ \mathcal{M}_6(0) \ | + | $$ \mathcal{M}_6(0) \equiv\mathcal{M}^{\Top}_6(0).$$ |
{{endthm}} | {{endthm}} | ||
Revision as of 12:45, 27 November 2010
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
Tex syntax errorbe the set of diffeomorphism classes of closed smooth simply-connected 2-connected 6-manifolds (the notation is used to be consistent with 6-manifolds: 1-connected). The classification
Tex syntax errorwas one of Smale's first applications of the h-cobordism theorem [Smale1962a, Corollary 1.3]. The is a precise 6-dimensional analogue of the classification of orientable surfaces: every 2-connected 6-manifold is diffeomorphic to a connected-sum
where by definition and in general is determined by the formula for the Euler characteristic of
Tex syntax error
Tex syntax error, see 6-manifolds: 1-connected.
2 Construction and examples
The following gives a complete list of 2-connected 6-manifolds up to diffeomorphism:
-
Tex syntax error
, the standard 6-sphere. - , the -fold connected sum of
Tex syntax error
.
3 Invariants
Suppose that is diffeomorphic to then:
-
Tex syntax error
, - the third Betti-number of is given by
Tex syntax error
, - the Euler characteristic of is given by
Tex syntax error
, - the intersection form of is isomorphic to the sum of b-copies of
Tex syntax error
, the standard skew-symmetric hyperbolic form on .
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 byTex syntax error.
Tex syntax errordenotes the natural numbers we obtain a bijection
Tex syntax error
5 Further discussion
5.1 Topological 2-connected 6-manifolds
Let 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. In particular, there is a bijection
Proof.
For any such manifold we haveTex syntax errorand 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
Let denote the group of isotopy classes of diffeomorphisms of a -connected -manifold and let denote the group of isomorphisms of perserving the intersection form: is the symplectic group when . By [Cerf1970] the forgetful map to the group of orientation preserving pseudo-isotopy classes of is an isomorphism. Based of Cerf's theorem in[Kreck1979] on finds exact sequences
where by definition is the subgroup of isotopy classes induced the identity on , and is the group of homotopy -spheres.
In particular .
For more information about the extension above, see [Krylov2003], [Johnson1983] and [Crowley2009].
References
- [Cerf1970] J. Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, Inst. Hautes Études Sci. Publ. Math. (1970), no.39, 5–173. MR0292089 (45 #1176) Zbl 0213.25202
- [Crowley2009] D. Crowley, On the mapping class groups of for , (2009). Available at the arXiv:0905.0423.
- [Johnson1983] D. Johnson, A survey of the Torelli group, Low-dimensional topology (San Francisco, Calif., 1981), Amer. Math. Soc. (1983), 165–179. MR718141 (85d:57009) Zbl 0553.57002
- [Kreck1979] M. Kreck, Isotopy classes of diffeomorphisms of -connected almost-parallelizable -manifolds, Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), Springer (1979), 643–663. MR561244 (81i:57029) Zbl 0421.57009
- [Krylov2003] N. A. Krylov, On the Jacobi group and the mapping class group of , Trans. Amer. Math. Soc. 355 (2003), no.1, 99–117 (electronic). MR1928079 (2003i:57039) Zbl 1015.57020
- [Smale1962a] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR0153022 (27 #2991) Zbl 0109.41103