6-manifolds: 1-connected
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== Introduction == | == Introduction == | ||
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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 Smale, extended by Wall and Jupp and completed by 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 Smale, extended by Wall and Jupp and completed by Zhubr. We shall write $\mathcal{M}^{\Cat}_{6}(e)$ for either $\mathcal{M}^{}_{6}(e)$ or $\mathcal{M}^{\Top}_{6}(e)$. | ||
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== Examples and constructions == | == Examples and constructions == | ||
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We first present some familiar 6-manifolds. | We first present some familiar 6-manifolds. | ||
* $S^6$, the standard 6-sphere. | * $S^6$, the standard 6-sphere. | ||
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* ??? Complete intersections of some form. | * ??? Complete intersections of some form. | ||
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== Classification == | == Classification == | ||
=== Smoothing theory === | === Smoothing theory === | ||
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{{beginthm|Theorem 1}} | {{beginthm|Theorem 1}} | ||
Let $M$ be a simply-connected, topological 6-manifold. The [[wikipedia:Kirby–Siebenmann_class|Kirby-Siebenmann class]], $\KS(M) \in H^4(M; \Zz_2)$ is the sole obstruction to $M$ admitting a smooth structure. | Let $M$ be a simply-connected, topological 6-manifold. The [[wikipedia:Kirby–Siebenmann_class|Kirby-Siebenmann class]], $\KS(M) \in H^4(M; \Zz_2)$ is the sole obstruction to $M$ admitting a smooth structure. | ||
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$$ \mathcal{M}_6(e) \rightarrow \mathcal{M}^{\Top}_6(e).$$ | $$ \mathcal{M}_6(e) \rightarrow \mathcal{M}^{\Top}_6(e).$$ | ||
{{endthm}} | {{endthm}} | ||
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=== The second Stiefel-Whitney class === | === The second Stiefel-Whitney class === | ||
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The second [[wikipedia:Stiefel–Whitney_class|Stiefel-Whitney class]] of $M$ is an element of $H^2(M; \Zz_2)$ which we regard as a homomorphism $w\co H_2(M) \rightarrow \Zz_2$. Let $\Hom({\mathcal Ab}, \Zz_2)$ be the set of isomorphism classes of pairs $(G, \omega)$ where $G$ is a finitely generated abelian group $w\co G \rightarrow \Zz_2$ is a homomorphism and where an isomorphism is an isomorphism of groups commuting with the homomorphisms to $\Zz_2$. The second Stiefel-Whitney classes defines a surjection | The second [[wikipedia:Stiefel–Whitney_class|Stiefel-Whitney class]] of $M$ is an element of $H^2(M; \Zz_2)$ which we regard as a homomorphism $w\co H_2(M) \rightarrow \Zz_2$. Let $\Hom({\mathcal Ab}, \Zz_2)$ be the set of isomorphism classes of pairs $(G, \omega)$ where $G$ is a finitely generated abelian group $w\co G \rightarrow \Zz_2$ is a homomorphism and where an isomorphism is an isomorphism of groups commuting with the homomorphisms to $\Zz_2$. The second Stiefel-Whitney classes defines a surjection | ||
$$ w\co\mathcal{M}_{6}^{\Top}(e) \rightarrow \Hom({\mathcal Ab}, \Zz_2)$$ | $$ w\co\mathcal{M}_{6}^{\Top}(e) \rightarrow \Hom({\mathcal Ab}, \Zz_2)$$ | ||
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$$ \mathcal{M}^{\Cat}_{6}(e) = \cup_{[G, w]} \mathcal{M}^{\Cat}_{6}(G, w)$$ | $$ \mathcal{M}^{\Cat}_{6}(e) = \cup_{[G, w]} \mathcal{M}^{\Cat}_{6}(G, w)$$ | ||
where $[G, w]$ ranges over all of $\Hom({\mathcal Ab}, \Zz_2)$. | where $[G, w]$ ranges over all of $\Hom({\mathcal Ab}, \Zz_2)$. | ||
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== 2-connected 6-manifolds == | == 2-connected 6-manifolds == | ||
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Smale showed that every smooth, 2-connected 6-manifold is diffeomorphic to $S^6$ or a connected sum $\sharp_r(S^3 \times S^3)$. Hence if $b_3(M)$ denotes the third Betti-number of $M$ and $\Nn$ denotes the natural numbers we obtain a bijection | Smale showed that every smooth, 2-connected 6-manifold is diffeomorphic to $S^6$ or a connected sum $\sharp_r(S^3 \times S^3)$. Hence if $b_3(M)$ denotes the third Betti-number of $M$ and $\Nn$ denotes the natural numbers we obtain a bijection | ||
$$ \mathcal{M}_6(0)\equiv \Nn,~~~[M] \mapsto \frac{1}{2}b_3(M).$$ | $$ \mathcal{M}_6(0)\equiv \Nn,~~~[M] \mapsto \frac{1}{2}b_3(M).$$ | ||
Applying Theorems 1 and 2 we see that the same statement holds for $\mathcal{M}^{\Top}_6(0)$. | Applying Theorems 1 and 2 we see that the same statement holds for $\mathcal{M}^{\Top}_6(0)$. | ||
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=== The splitting Theorem === | === The splitting Theorem === | ||
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{{beginthm|Theorem 3|(Wall)}} | {{beginthm|Theorem 3|(Wall)}} | ||
Let $M$ be a closed, smooth, simply-connected 6-manifold with $b_3(M) = 2r$. Then up to diffeomorphism, there is a unique maniofld $M_0$ with $b_3(M_0) = 0$ such that $M$ is diffeomorphic to $M_0 \sharp_r(S^3 \times S^3)$. | Let $M$ be a closed, smooth, simply-connected 6-manifold with $b_3(M) = 2r$. Then up to diffeomorphism, there is a unique maniofld $M_0$ with $b_3(M_0) = 0$ such that $M$ is diffeomorphic to $M_0 \sharp_r(S^3 \times S^3)$. | ||
{{endthm}} | {{endthm}} | ||
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== 6-manifolds with torsion free second homology == | == 6-manifolds with torsion free second homology == | ||
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In addition to $w\co H_2(M) \rightarrow \Zz_2$ and $b_3(M)$ we require the following invariants: | In addition to $w\co H_2(M) \rightarrow \Zz_2$ and $b_3(M)$ we require the following invariants: | ||
* The first [[wikipedia:Pontrjagin_class|Pontrjagin class]] $p_1(M) \in H^4(M)$. | * The first [[wikipedia:Pontrjagin_class|Pontrjagin class]] $p_1(M) \in H^4(M)$. | ||
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$$W^3 = (p_1(M) + 24K) \cup W$$ | $$W^3 = (p_1(M) + 24K) \cup W$$ | ||
for all $W \in H^2(M)$ which reduce to $w_2(M)$ mod $2$ and for all $K \in H^4(M)$ which reduce to $\KS(M)$ mod $2$. | for all $W \in H^2(M)$ which reduce to $w_2(M)$ mod $2$ and for all $K \in H^4(M)$ which reduce to $\KS(M)$ mod $2$. | ||
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== References == | == References == |
Revision as of 12:04, 28 August 2009
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 Smale, extended by Wall and Jupp and completed by Zhubr. We shall write for either or .
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 Classification
1.1 Smoothing theory
Theorem 1 5.1. Let be a simply-connected, topological 6-manifold. The Kirby-Siebenmann class, is the sole obstruction to admitting a smooth structure.
Theorem 2 5.2. Every homeomorphism of simply-connected, smooth -manifolds is topologically isotopic to a diffeomorphism. Hence we have an injection
2.1 The second Stiefel-Whitney class
Tex syntax error. 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 .
3 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 .
3.1 The splitting Theorem
Theorem 3 6.1 (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 .
4 6-manifolds with torsion free second homology
In addition to and we require the following invariants:
- 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 .
5 References
- [Jupp1973] P. E. Jupp, Classification of certain -manifolds, Proc. Cambridge Philos. Soc. 73 (1973), 293–300. MR0314074 (47 #2626) Zbl 0249.57005
- [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 0149.20601MediaWiki:Stub