6-manifolds: 2-connected
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
[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
![\displaystyle M \cong \#_r(S^3 \times S^3)](/images/math/a/1/9/a19f19e89966cbaacbe2f803cd5350f8.png)
where by definition and in general
is determined by the formula for the Euler characteristic of
![\displaystyle \chi(M) = 2 - 2r./var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_DdcQSa](/images/math/9/c/9/9c9cd92233e15d5523da63c3fd1322d0.png)
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
![\displaystyle \mathcal{M}_6(0)\equiv \Nn,~~~[M] \mapsto \frac{1}{2}b_3(M)./var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_4Uv0PL](/images/math/4/f/8/4f87caab28b3b9b7fd16a5f88c9bb677.png)
[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
![\displaystyle \mathcal{M}_6(0) \equiv\mathcal{M}^{\Top}_6(0).](/images/math/8/e/4/8e46f712bf459c035223fcccc69d581d.png)
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.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
[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
![\displaystyle 0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\Diff_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0 \quad (\ast)](/images/math/d/f/0/df015be610eaa218f9a45d256eccdcdf.png)
![\displaystyle 0 \rightarrow \Theta_7 \rightarrow \pi_0\SDiff(M) \rightarrow H^3(M) \rightarrow 0](/images/math/6/a/5/6a5be654e30ac9b8adf8497175d6ace3.png)
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
![\displaystyle \alpha_b : B\mathcal{D}_b \longrightarrow \Omega^\infty_\bullet MT\mathrm{Spin}(6)](/images/math/d/1/0/d107edc6a8f79989e7fb9f9cd4c7d158.png)
to the basepoint component of the infinite loop space of the spectrum , and these fit together under the maps
to give a map
![\displaystyle \alpha : \mathrm{hocolim}_{b \to \infty} B\mathcal{D}_b \longrightarrow \Omega^\infty_\bullet MT\mathrm{Spin}(6).](/images/math/5/4/a/54aa6a6a79d9d4f95e7e9be7176bad79.png)
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