Topology Seminar

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)