Talk:5-manifolds: 1-connected
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Revision as of 13:34, 16 December 2010 by Diarmuid Crowley (Talk | contribs)
1 Conjecture about mapping class groups of 1-connected 5-manifolds
Let 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))](/images/math/f/4/0/f4098ae63acd5841fa91e600a08f530e.png)
where is the normal
-type of
as defined in [Kreck1999]. For example, if
is Spinable with
then
![\displaystyle \Omega_6(B^2(M)) \cong \Omega_6^{Spin}(K(H, 2)).](/images/math/4/6/1/461cca1153f63aa9c27b6fdcbd9d014c.png)
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)
2 Earlier work of Fang
The group was computed in [Fang1993] provided that
has no
-torsion and no
-torsion.
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)](/images/math/a/4/7/a474064900f2f9c81fa306dbdd7c5d9f.png)
we obtain an action of on the abelian group
. Diarmuid Crowley and Matthias Kreck also conjecture that the action of
is via the induced action on
.