# Talk:5-manifolds: 1-connected/1st edition

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## Conjecture about mapping class groups of 1-connected 5-manifolds

Let $M$$== Conjecture about mapping class groups of 1-connected 5-manifolds == ; Let M be a closed, smooth, 1-connected 5-manifold, [[User:Matthias Kreck|Matthias Kreck]] and [[User:Diarmuid Crowley|Diarmuid Crowley]] conjecture that there is an isomorphism of abelian groups \pi_0(\SDiff(M)) \cong \Omega_6(B_2(M)) where B_2(M) is the normal -type of M as defined in {{cite|Kreck1999}}. For example, if M is Spinable with H_2(M) \cong H then \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 {{cite|Kreck1999}}. [[User:Diarmuid Crowley|Diarmuid Crowley]] 10:02, 29 September 2009 (UTC)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)$$B_2(M)$ is the normal $2$$2$-type of $M$$M$ as defined in [Kreck1999]. For example, if $M$$M$ is Spinable with $H_2(M) \cong H$$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)