Talk:5-manifolds: 1-connected

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[edit] 1 Conjecture about mapping class groups of 1-connected 5-manifolds

Let M be a closed, smooth, 1-connected 5-manifold, Matthias Kreck and Diarmuid Crowley conjecture that there is an isomorphism of abelian groups

\displaystyle  \pi_0(\SDiff(M)) \cong \Omega_6(B^2(M))

where B^2(M) is the normal 2-type of M as defined in [Kreck1999]. For example, if M is Spinable with H_2(M) \cong H then

\displaystyle  \Omega_6(B^2(M)) \cong \Omega_6^{\Spin}(K(H, 2)).

At present we are checking the details of the proof of this conjecture using the methods of [Kreck1999].

Diarmuid Crowley 10:02, 29 September 2009 (UTC)

[edit] 2 Earlier work of Fang

The group \pi_0\SDiff(M) was computed in [Fang1993] provided that
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has no 2-torsion and no 3-torsion.

[edit] Up-date of conjecture: module structure

If the conjecture above holds, then from the short exact sequence

\displaystyle  0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\text{Diff}_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0 \quad (\ast)

we obtain an action of \Aut(H_2(M)) on the abelian group \pi_0\SDiff(M) \cong \Omega_6(B^2(M)). Diarmuid Crowley and Matthias Kreck also conjecture that the action of \Aut(H_2(M)) is via the induced action on B^2(M). In particular, if M is spinable with H = H_2(M), then \Aut(H_2(M)) acts on K(H, 2) in the obvious way and so on \Omega_6^{\Spin}(K(H, 2)).

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