6-manifolds: 1-connected
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Let $\mathcal{M}_{6}(e)$ be the set of diffeomorphism classes of [[wikipedia:Closed_manifold|closed]], [[wikipedia:Oriented_manifold#Orientability_of_manifolds|oriented]], [[wikipedia:Differentiable_manifold|smooth]], [[wikipedia:Simply-connected|simply-connected]] 6-manifolds $M$. Similarly, let $\mathcal{M}^{\Top}_{6}(e)$ be the set of [[wikipedia:Homeomorphism|homeomorphism]] classes of closed, oriented [[wikipedia:Topological_manifold|topological manifolds]]. | Let $\mathcal{M}_{6}(e)$ be the set of diffeomorphism classes of [[wikipedia:Closed_manifold|closed]], [[wikipedia:Oriented_manifold#Orientability_of_manifolds|oriented]], [[wikipedia:Differentiable_manifold|smooth]], [[wikipedia:Simply-connected|simply-connected]] 6-manifolds $M$. Similarly, let $\mathcal{M}^{\Top}_{6}(e)$ be the set of [[wikipedia:Homeomorphism|homeomorphism]] classes of closed, oriented [[wikipedia:Topological_manifold|topological manifolds]]. | ||
− | In this article we report on the calculation of $\mathcal{M}^{\Cat}_{6}(e)$ and $\mathcal{M}^{\Top}_{6}(e)$ begun by {{cite|Smale1962}}, extended in {{cite|Wall1966}} in {{cite|Jupp1973}} and completed in {{cite|Zhubr}}. We shall write $\mathcal{M}^{\Cat}_{6}(e)$ for either $\mathcal{M}^{}_{6}(e)$ or $\mathcal{M}^{\Top}_{6}(e)$. | + | In this article we report on the calculation of $\mathcal{M}^{\Cat}_{6}(e)$ and $\mathcal{M}^{\Top}_{6}(e)$ begun by {{cite|Smale1962}}, extended in {{cite|Wall1966}} in {{cite|Jupp1973}} and completed in {{cite|Zhubr}}. We shall write $\mathcal{M}^{\Cat}_{6}(e)$ for either $\mathcal{M}^{}_{6}(e)$ or $\mathcal{M}^{\Top}_{6}(e)$. An excellent summary for the case where $H_2(M)$ is torsion free may be found in {{cite|Okonek&Van de Ven1995}}. |
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Revision as of 16:11, 7 June 2010
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Contents |
1 Introduction
Let be the set of diffeomorphism classes of closed, oriented, smooth, simply-connected 6-manifolds . Similarly, let be the set of homeomorphism classes of closed, oriented topological manifolds. In this article we report on the calculation of and begun by [Smale1962], extended in [Wall1966] in [Jupp1973] and completed in [Zhubr]. We shall write for either or . An excellent summary for the case where is torsion free may be found in [Okonek&Van de Ven1995].
2 Examples and constructions
We first present some familiar 6-manifolds.
- , the standard 6-sphere.
- , the -fold connected sum of .
- , the -fold connected sum of .
- , 3-dimensional complex projective space.
- , the non-trivial linear 4-sphere bundle over .
- For each we have , the corresponding 2-sphere bundle over . If we write 1 for a generator of then is diffeomorphic to .
Surgery on framed links. Let be a framed link. Then , the outcome of surgery on , is a simply connected Spinable 6-manifold with and .
- ??? Complete intersections of some form.
1 Invariants
The second Stiefel-Whitney class of is an element of which we regard as a homomorphism .
- The first Pontrjagin class .
- The Kirby-Siebenmann class
- The cup product .
These invariants satisfy the following relation
for all which reduce to mod and for all which reduce to mod .
2 Classification
2.1 Preliminaries
Let be the set of isomorphism classes of pairs where is a finitely generated abelian group is a homomorphism and where an isomorphism is an isomorphism of groups commuting with the homomorphisms to . The second Stiefel-Whitney classes defines a surjection
and we let denote the set of isomorphism classes of 6-manifolds with prescribed second Stiefel-Whitney class. We obtain the decomposition
where ranges over all of .
2.1 The splitting Theorem
Theorem 3 8.1 (Wall). Let be a closed, smooth, simply-connected 6-manifold with . Then up to diffeomorphism, there is a unique maniofld with such that is diffeomorphic to .
2.2 Smoothing theory
Theorem 1 8.2. Let be a simply-connected, topological 6-manifold. The Kirby-Siebenmann class, is the sole obstruction to admitting a smooth structure.
Theorem 2 8.3. Every homeomorphism of simply-connected, smooth -manifolds is topologically isotopic to a diffeomorphism. Hence we have an injection
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3 6-manifolds with torsion free second homology
4 2-connected 6-manifolds
Smale showed that every smooth, 2-connected 6-manifold is diffeomorphic to or a connected sum . Hence if denotes the third Betti-number of and denotes the natural numbers we obtain a bijection
Applying Theorems 1 and 2 we see that the same statement holds for .
5 Further discussion
6 References
- [Jupp1973] P. E. Jupp, Classification of certain -manifolds, Proc. Cambridge Philos. Soc. 73 (1973), 293–300. MR0314074 (47 #2626) Zbl 0249.57005
- [Okonek&Van de Ven1995] C. Okonek and A. Van de Ven, Cubic forms and complex -folds, Enseign. Math. (2) 41 (1995), no.3-4, 297–333. MR1365849 (97b:32035) Zbl 0869.14018
- [Smale1962] S. Smale, On the structure of -manifolds, Ann. of Math. (2) 75 (1962), 38–46. MR0141133 (25 #4544) Zbl 0101.16103
- [Wall1966] C. T. C. Wall, Classification problems in differential topology. V. On certain -manifolds, Invent. Math. 1 (1966), 355-374; corrigendum, ibid 2 (1966), 306. MR0215313 (35 #6154) Zbl 0149.20601
- [Zhubr] Template:Zhubr