# Talk:5-manifolds: 1-connected

## [edit] 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

where is the normal -type of as defined in [Kreck1999]. For example, if is Spinable with then

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 was computed in [Fang1993] provided that has no -torsion and no -torsion.

## [edit] Up-date of conjecture: module structure

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

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 . In particular, if is spinable with , then acts on in the obvious way and so on .