# Talk:5-manifolds: 1-connected

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## 1 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) == Earlier work of Fang == ; The group \pi_0\SDiff(M) was computed in \cite{Fang1993} provided that H_2(M) has no -torsion and no -torsion. == Up-date of conjecture: module structure == ; If the conjecture above holds, then from the short exact sequence 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)). [[User:Diarmuid Crowley|Diarmuid Crowley]] and [[User:Matthias Kreck|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)). 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)

## 2 Earlier work of Fang

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

##  Up-date of conjecture: module structure

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

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

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