Talk:5-manifolds: 1-connected
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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 | 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 | ||
− | $$ \pi_0(\SDiff(M)) \cong \Omega_6( | + | $$ \pi_0(\SDiff(M)) \cong \Omega_6(B^2(M)) $$ |
− | where $ | + | where $B^2(M)$ is the normal $2$-type of $M$ as defined in {{cite|Kreck1999}}. For example, if $M$ is Spinable with $H_2(M) \cong H$ then |
− | $$ \Omega_6( | + | $$ \Omega_6(B^2(M)) \cong \Omega_6^{\Spin}(K(H, 2)).$$ |
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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== Earlier work of Fang == | == Earlier work of Fang == | ||
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− | The group $\pi_0\SDiff(M)$ was computed in \cite{Fang1993} provided that $H_2(M)$ has no $2$-torsion and no $3$-torsion. | + | The group $\pi_0\SDiff(M)$ was computed in \cite{Fang1993} provided that $H_2(M)$ has no $2$-torsion and no $3$-torsion. |
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+ | == Up-date of conjecture: module structure == | ||
+ | <wikitex>; | ||
+ | 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 $\Omega_6^{\Spin}(K(H, 2))$. | ||
+ | </wikitex> |
Latest revision as of 19:05, 16 December 2010
[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 .