Timeline for Mappings of mapping class groups
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
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| Dec 3, 2009 at 4:00 | comment | added | Ryan Budney | Rereading Richard's post I see my comment really should have $n=4$ for the torus and $n=6$ for the genus 2 surface. if $n<4$ for the torus or $n<6$ for the genus two surface there may not be a splitting since the mapping class group permutes the fixed points of the hyper-elliptic involution. | |
| Dec 3, 2009 at 3:27 | comment | added | Ryan Budney | Ah, sorry about misreading your question. The map $MCG(M,n) \to MCG(M)$ has a splitting in several natural cases -- for the torus if $n \leq 4$, and the genus 2 surface if $n \leq 6$. This is because of the Dehn twist generators in these cases commute with the hyper-elliptic involution of the surface. In general I think that map is non-split -- sometimes $MCG(M)$ has bigger torsion subgroups than $MCG(M,n)$. But in general I don't know an obstruction for (or a construction of) an embedding $MCG(g,0) \to MCG(g',n)$ for $n>0$. | |
| Dec 3, 2009 at 3:11 | comment | added | Greg Kuperberg |
In 3, he's asking whether the extension by $\pi_1$ splits.
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| Dec 3, 2009 at 2:40 | comment | added | algori | Ryan -- thanks! Will take a look at Birman-Hilden's paper tomorrow. Re the last question: the above sequence does not seem to help much, since I'm looking for a map in the other direction, i.e. from the MCG of a surface with no boundary to the MCG of a surface with boundary. And yes, there is a reason for that: I am trying to prove something about certain representations of mapping class groups. The case when there is no boundary is more difficult, as usual, I imagine. But it would follow from the easier case when there is a boundary if there is an injection $MCG(g,0)\to MCG (g',n)$. | |
| Dec 3, 2009 at 2:27 | comment | added | HJRW | I read part 3 the other way round: it asks whether there are any embeddings of MCG(X) into MCG(Y,n) for any X, Y, n. | |
| Dec 3, 2009 at 1:53 | history | answered | Ryan Budney | CC BY-SA 2.5 |