6-manifolds: 2-connected

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[edit] 1 Introduction

Let \mathcal{M}_6(0) be the set of diffeomorphism classes of closed smooth simply-connected 2-connected 6-manifolds M (the notation is used to be consistent with 6-manifolds: 1-connected).

The classification \mathcal{M}_6(0) 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 M is diffeomorphic to a connected-sum

\displaystyle  M \cong \#_r(S^3 \times S^3)

where by definition \#_0(S^3 \times S^3) = S^6 and in general r is determined by the formula for the Euler characteristic of M

\displaystyle  \chi(M) = 2 - 2r.

For the more general case where H_2(M) \neq 0, see 6-manifolds: 1-connected.

[edit] 2 Construction and examples

The following gives a complete list of 2-connected 6-manifolds up to diffeomorphism:

  • S^6, the standard 6-sphere.
  • \#_b(S^3 \times S^3), the b-fold connected sum of S^3 \times S^3.

[edit] 3 Invariants

Suppose that M is diffeomorphic to \#_b(S^3 \times S^3) then:

  • \pi_3(M) \cong H_3(M) \cong \Zz^{2b},
  • the third Betti-number of M is given by b_3(M) = 2b,
  • the Euler characteristic of M is given by \chi(M) = 2 - 2b,
  • the intersection form of M is isomorphic to the sum of b-copies of H_{-}(\Zz), the standard skew-symmetric hyperbolic form on \Zz^2.

[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 S^3 \times S^3.

Hence if \Nn denotes the natural numbers we obtain a bijection

\displaystyle  \mathcal{M}_6(0)\equiv \Nn,~~~[M] \mapsto \frac{1}{2}b_3(M).

[edit] 5 Further discussion

[edit] 5.1 Topological 2-connected 6-manifolds

Let \mathcal{M}^{\Top}_6(0) 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).

Proof. For any such manifold M we have H^4(M; \Zz/2) \cong 0 and so M 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 \pi_0\Diff_+(M) denote the group of isotopy classes of diffeomorphisms f \colon M \to M of a 2-connected 6-manifold M and let \Aut(M) denote the group of isomorphisms of H_3(M) perserving the intersection form: \Aut(M) \cong Sp_{2b}(\Zz) is the symplectic group when M = \#_b(S^3 \times S^3). By [Cerf1970] the forgetful map to the group of orientation preserving pseudo-isotopy classes of M 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)
\displaystyle  0 \rightarrow \Theta_7 \rightarrow \pi_0\SDiff(M) \rightarrow H^3(M) \rightarrow 0

where by definition \pi_0\SDiff(M) is the subgroup of isotopy classes induced the identity on H_*(M) and \Theta_7 \cong \pi_0(\Diff(D^6, \partial)) is the group of homotopy 7-spheres.

In particular \pi_0(\Diff_+(S^6)) \cong \Zz/28 \cong \Theta_7.

For more information about the extensions in (\ast) above, see [Krylov2003], [Johnson1983] and [Crowley2009].

[edit] 5.3 Diffeomorphism groups

Let \mathcal{D}_b = \Diff(\#_b S^3 \times S^3, D^6) denote group of diffeomorphisms of \#_b S^3 \times S^3 which are the identity inside a marked disc, and B\mathcal{D}_b denote the classifying spaces of this topological group. Connect-sum inside the marked disc gives homomorphisms \mathcal{D}_b \to \mathcal{D}_{b+1}, and so continuous maps \mathcal{S} : B\mathcal{D}_b \to B\mathcal{D}_{b+1}. The homology of these classifying spaces is approachable in a range of degrees, by the following theorem.

Theorem 5.2 [Galatius&Randal-Williams2012, Theorem 1.2]. The map

\displaystyle \mathcal{S}_* : H_k(B\mathcal{D}_b) \longrightarrow H_k(B\mathcal{D}_{b+1})

is an isomorphism for b \geq 2k+4.

Let B\mathrm{Spin}(6) denote the classifying space of the group \mathrm{Spin}(6), and \gamma_6^{\mathrm{Spin}} denote its universal vector bundle. We write MT\mathrm{Spin}(6) for the Thom spectrum of the virtual bundle -\gamma_6^{\mathrm{Spin}}. Pontrjagin--Thom theory provides a map

\displaystyle \alpha_b : B\mathcal{D}_b \longrightarrow \Omega^\infty_\bullet MT\mathrm{Spin}(6)

to the basepoint component of the infinite loop space of the spectrum MT\mathrm{Spin}(6), and these fit together under the maps \mathcal{S} to give a map

\displaystyle \alpha : \mathrm{hocolim}_{b \to \infty} B\mathcal{D}_b \longrightarrow \Omega^\infty_\bullet MT\mathrm{Spin}(6).

Theorem 5.3 [Galatius&Randal-Williams2012a, Corollary 1.2]. The map \alpha induces an isomorphism on (co)homology.

It is not difficult to calculate the rational cohomology of \Omega^\infty_\bullet MT\mathrm{Spin}(6), and find that it is a polynomial algebra with generators in degrees 2,2,4,6,6,6,8,8,10,10,10,12,12,\ldots, 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 B\mathcal{D}_b in degrees * \leq (b-4)/2.

[edit] 6 References

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