Talk:5-manifolds: 1-connected
(Difference between revisions)
m |
m |
||
Line 19: | Line 19: | ||
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: | + | 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 $B^2(M)$. |
</wikitex> | </wikitex> |
Revision as of 12:35, 16 December 2010
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)
2 Earlier work of Fang
Tex syntax errorhas no -torsion and no -torsion.
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 .