6-manifolds: 1-connected
m (moved Simply-connected 6-manifolds to 6-manifolds: 1-connected: To start names with dimension) |
|||
Line 20: | Line 20: | ||
* ??? Complete intersections of some form. | * ??? Complete intersections of some form. | ||
<wikitex>; | <wikitex>; | ||
+ | |||
+ | == Invariants == | ||
+ | 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 $w\co H_2(M) \rightarrow \Zz_2$. | ||
+ | |||
+ | * The first [[wikipedia:Pontrjagin_class|Pontrjagin class]] $p_1(M) \in H^4(M)$. | ||
+ | * The Kirby-Siebenmann class $\KS(M) \in H^4(M; \Zz_2)$ | ||
+ | * The cup product $\cup_3\co H^2(M) \otimes H^2(M) \otimes H^2(M) \rightarrow H^6(M) = \Zz$. | ||
+ | These invariants satisfy the following relation | ||
+ | $$W^3 = (p_1(M) + 24K) \cup W$$ | ||
+ | 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 $2$. | ||
== Classification == | == Classification == | ||
+ | |||
+ | === Preliminaries === | ||
+ | <wikitex>; | ||
+ | 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 | ||
+ | $$ w\co\mathcal{M}_{6}^{\Top}(e) \rightarrow \Hom({\mathcal Ab}, \Zz_2)$$ | ||
+ | and we let $\mathcal{M}^{\Cat}_6(G, w) = w^{-1}([G, w])$ denote the set of isomorphism classes of 6-manifolds with prescribed second Stiefel-Whitney class. We obtain the decomposition | ||
+ | $$ \mathcal{M}^{\Cat}_{6}(e) = \cup_{[G, w]} \mathcal{M}^{\Cat}_{6}(G, w)$$ | ||
+ | where $[G, w]$ ranges over all of $\Hom({\mathcal Ab}, \Zz_2)$. | ||
+ | </wikitex> | ||
+ | |||
+ | === The splitting Theorem === | ||
+ | <wikitex>; | ||
+ | {{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 $M_0 \sharp_r(S^3 \times S^3)$. | ||
+ | {{endthm}} | ||
+ | </wikitex> | ||
+ | |||
=== Smoothing theory === | === Smoothing theory === | ||
<wikitex>; | <wikitex>; | ||
Line 34: | Line 61: | ||
</wikitex> | </wikitex> | ||
− | == | + | == 6-manifolds with torsion free second homology == |
<wikitex>; | <wikitex>; | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
</wikitex> | </wikitex> | ||
Line 50: | Line 73: | ||
</wikitex> | </wikitex> | ||
− | == | + | == Further discussion == |
<wikitex>; | <wikitex>; | ||
− | + | </nowikitex> | |
− | + | ||
− | + | ||
− | </ | + | |
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
== References == | == References == |
Revision as of 15:55, 7 June 2010
This page has not been refereed. The information given here might be incomplete or provisional. |
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
Tex syntax error
. - , 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
Tex syntax error
3 6-manifolds with torsion free second homology
4 2-connected 6-manifolds
Tex syntax errordenotes the natural numbers we obtain a bijection
Tex syntax error
Applying Theorems 1 and 2 we see that the same statement holds for .
5 Further discussion
</nowikitex>
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 0149.20601