Talk:5-manifolds: 1-connected

Revision as of 14:05, 29 October 2010

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

Let $== Conjecture about mapping class groups of 1-connected 5-manifolds == <wikitex>; 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}}. </wikitex> [[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 $B_2(M)$ 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)

2 Earlier work of Fang

The group $\pi_0\SDiff(M)$ was computed in [Fang1993] provided that $H_2(M)$ has no $2$-torsion and no $3$-torsion. ~~~~