Abstract: This is joint work with Jeremy Miller and Jennifer Wilson. Let M be an orientable surface of genus g with one boundary curve, and let F_n(M) denote the configuration space of n ordered points in M. The action of Homeo(M,dM) on F_n(M) descends to an action of the mapping class group Gamma(M,dM) on the homology H_*(F_n(M)). Our main result is that, for all n,i>=0, the i-th stage J(i) of the Johnson’s filtration of Gamma(M,dM) acts trivially on H_i(F_n(M)). This extends previous work of Moriyama on certain relative configuration spaces.
I will recall the necessary definitions and give a sketch of the proof of the main theorem: the main inputs are Moriyama’s work and a cell stratification of F_n(M) à la Fox-Neuwirth-Fuchs. I will also present some examples of non-trivial actions of mapping classes in J(i-1) on elements of H_i(F_n(M)), for small values of i.