6-manifolds: 1-connected
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* $\sharp_r S^2 \times S^4$, the $r$-fold connected sum of $S^2 \times S^4$. | * $\sharp_r S^2 \times S^4$, the $r$-fold connected sum of $S^2 \times S^4$. | ||
* $\CP^3$, 3-dimensional complex projective space. | * $\CP^3$, 3-dimensional complex projective space. | ||
− | * $S^ | + | * $S^4 \times_\gamma S^2$, the non-trivial linear 4-sphere bundle over $S^2$. |
− | * For each $\alpha \in \pi_3(\SO_3) \cong \Zz$ we have $S^ | + | * For each $\alpha \in \pi_3(\SO_3) \cong \Zz$ we have $S^2 \times_\alpha S^4$, the corresponding 2-sphere bundle over $S^4$. If we write 1 for a generator of $\pi_3(\SO_3)$ then $S^2 \times_1 S^4$ is diffeomorphic to $\CP^3$. |
Surgery on framed links. Let $\phi \co \sqcup_r S^3 \times D^3 \to S^6$ be a framed link. Then $M^6_\phi$, the outcome of surgery on $\phi$, is a simply connected Spinable 6-manifold with $H_2(M_\phi) \cong H_4(M_\phi) \cong \Zz^r$ and $H_3(M_\phi) = 0$. | Surgery on framed links. Let $\phi \co \sqcup_r S^3 \times D^3 \to S^6$ be a framed link. Then $M^6_\phi$, the outcome of surgery on $\phi$, is a simply connected Spinable 6-manifold with $H_2(M_\phi) \cong H_4(M_\phi) \cong \Zz^r$ and $H_3(M_\phi) = 0$. |
Revision as of 17:17, 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 [Zhubr2000]. 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.
3 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 . As [Okonek&Van de Ven1995, p. 300] remark, in the smooth case this follows from the integrality of the -genus but in the topological case requires further arguments carried out in [Jupp1973].
4 Classification
4.1 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 .
4.2 The splitting Theorem
Theorem 3 4.1 (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 .
4.3 Smoothing theory
Theorem 1 4.2. Let be a simply-connected, topological 6-manifold. The Kirby-Siebenmann class, is the sole obstruction to admitting a smooth structure.
Theorem 2 4.3. Every homeomorphism of simply-connected, smooth -manifolds is topologically isotopic to a diffeomorphism. Hence we have an injection
5 6-manifolds with torsion free second homology
...
6 Further discussion
...
7 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
- [Zhubr2000] A. V. Zhubr, Closed simply connected six-dimensional manifolds: proofs of classification theorems, Algebra i Analiz 12 (2000), no.4, 126–230. MR1793619 (2001j:57041)