TY - JOUR
T1 - The mapping class group action on the homology of the configuration spaces of surfaces
AU - Moriyama, Tetsuhiro
N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.
PY - 2007/10
Y1 - 2007/10
N2 - 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]).
AB - 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]).
UR - http://www.scopus.com/inward/record.url?scp=44649152407&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=44649152407&partnerID=8YFLogxK
U2 - 10.1112/jlms/jdm077
DO - 10.1112/jlms/jdm077
M3 - Article
AN - SCOPUS:44649152407
VL - 76
SP - 451
EP - 466
JO - Journal of the London Mathematical Society
JF - Journal of the London Mathematical Society
SN - 0024-6107
IS - 2
ER -