6-manifolds: 2-connected
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[edit] 1 Introduction
Let be 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 was 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
For the more general case where , see 6-manifolds: 1-connected.
[edit] 2 Construction and examples
The following gives a complete list of 2-connected 6-manifolds up to diffeomorphism:
- , the standard 6-sphere.
- , the -fold connected sum of .
[edit] 3 Invariants
Suppose that is diffeomorphic to then:
- ,
- the third Betti-number of is given by ,
- the Euler characteristic of is given by ,
- the intersection form of is isomorphic to the sum of b-copies of , the standard skew-symmetric hyperbolic form on .
[edit] 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 by .
Hence if denotes the natural numbers we obtain a bijection
[edit] 5 Further discussion
[edit] 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 have and 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.
[edit] 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. Applying Cerf's theorem Kreck proved in [Kreck1979] that there are 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 extensions in above, see [Krylov2003], [Johnson1983] and [Crowley2009].
[edit] 5.3 Diffeomorphism groups
Let denote group of diffeomorphisms of which are the identity inside a marked disc, and denote the classifying spaces of this topological group. Connect-sum inside the marked disc gives homomorphisms , and so continuous maps . The homology of these classifying spaces is approachable in a range of degrees, by the following theorem.
Let denote the classifying space of the group , and denote its universal vector bundle. We write for the Thom spectrum of the virtual bundle . Pontrjagin--Thom theory provides a map
to the basepoint component of the infinite loop space of the spectrum , and these fit together under the maps to give a map
Theorem 5.3 [Galatius&Randal-Williams2012a, Corollary 1.2]. The map induces an isomorphism on (co)homology.
It is not difficult to calculate the rational cohomology of , and find that it is a polynomial algebra with generators in degrees , which can be given an explicit description in terms of generalised Miller-Morita-Mumford classes. By the stability theorem, this calculates the rational cohomology of in degrees .
[edit] 6 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.
- [Galatius&Randal-Williams2012] S. Galatius and O. Randal-Williams, Homological stability for moduli spaces of high dimensional manifolds, (2012). Available at the arXiv:1203.6830.
- [Galatius&Randal-Williams2012a] S. Galatius and O. Randal-Williams, Stable moduli spaces of high dimensional manifolds, (2012). Available at the arXiv:1201.3527.
- [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