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Topology Seminar

Non-trivial action of the Johnson filtration on the homology of configuration spaces
Thursday, August 18, 2022
12:00-1:00 PM
4096 East Hall Map
This is joint work with Andreas Stavrou. Let S be a compact, connected, orientable surface, possibly with boundary, and let F_n(S) denote the space of ordered configurations of n distinct points in S. The homology groups H_*(F_n(S)) admit a natural action of the mapping class group Mod(S)=pi_0(Diff_+(S,dS)), and we are interested in how non-trivial this action is.
For S=S_{g,1}, we consider the Johnson filtration on Mod(S) by subgroups J(i), for i>=0. For g>=2, we prove that there are mapping classes in J(n-1) that act non-trivially on H_ (F_n(S)); this should be compared with a previous result of Miller, Wilson and myself, ensuring that each mapping class in the smaller subgroup J(n) acts trivially on H_(F_n(S)).
Similarly, for S=S_g, we consider an analogue of the Johnson filtration on Mod(S), and prove for g>=3 that J(n-1) admits elements acting non-trivially on H_*(F_{n+1}(S_g)).
I will sketch the proof of these statements, focusing on the construction of homology classes in H_*(F_n(S)). Speaker(s): Andrea Bianchi (University of Copenhagen)
Building: East Hall
Event Type: Workshop / Seminar
Tags: Mathematics
Source: Happening @ Michigan from Department of Mathematics