In this paper, we study the natural action of the mapping class group ℳg, 1 on the (co)homology groups of the configuration spaces of n-points on a surface Σ of genus g with the boundary ∂Σ ≅ S 1. We present two main results in this paper. The first result is that the kernel of the action of ℳg, 1 coincides with the kernel of the natural action on the nth lower central quotient group of the fundamental group of Σ. The second result is a new interpretation of the cohomology group H*(ℳg, 1; T[H1]) of ℳg, 1 with coefficients in the free tensor algebra T[H1] over generated by the first homology group H1 of Σ, by using the configuration spaces. More precisely, we define a certain cochain complex C of ℳg, 1-modules by using the configuration spaces and prove that H*(ℳg, 1; C) is canonically isomorphic to H*(ℳg, 1; T[H1]).
All Science Journal Classification (ASJC) codes