Talk:5-manifolds: 1-connected

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$$ \pi_0(\SDiff(M)) \cong \Omega_6(B^2(M)) $$
$$ \pi_0(\SDiff(M)) \cong \Omega_6(B^2(M)) $$
where $B_2(M)$ is the normal $2$-type of $M$ as defined in {{cite|Kreck1999}}. For example, if $M$ is Spinable with $H_2(M) \cong H$ then
where $B_2(M)$ is the normal $2$-type of $M$ as defined in {{cite|Kreck1999}}. For example, if $M$ is Spinable with $H_2(M) \cong H$ then
$$ \Omega_6(B^2(M)) \cong \Omega_6^{Spin}(K(H, 2)).$$
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$$ \Omega_6(B^2(M)) \cong \Omega_6^{\Spin}(K(H, 2)).$$
At present we are checking the details of the proof of this conjecture using the methods of {{cite|Kreck1999}}.
At present we are checking the details of the proof of this conjecture using the methods of {{cite|Kreck1999}}.
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If the conjecture above holds, then from the short exact sequence
If the conjecture above holds, then from the short exact sequence
$$ 0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\text{Diff}_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0 \quad (\ast)$$
$$ 0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\text{Diff}_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0 \quad (\ast)$$
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 $B^2(M)$.
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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 $\Omega_6^{\Spin}(K(H, 2))$.
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Revision as of 16:59, 16 December 2010

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

Let 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.



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)) on the abelian group \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)) 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)).

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