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The mapping class group action on the homology of the configuration spaces of surfaces
Author(s) -
Moriyama Tetsuhiro
Publication year - 2007
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdm077
Subject(s) - mathematics , homology (biology) , cohomology , group action , mapping class group , quotient , group (periodic table) , pure mathematics , combinatorics , group cohomology , kernel (algebra) , algebra over a field , surface (topology) , geometry , physics , quantum mechanics , biochemistry , chemistry , gene
In this paper, we study the natural action of the mapping class group ℳ g , 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 ℳ g , 1 coincides with the kernel of the natural action on the n th lower central quotient group of the fundamental group of Σ. The second result is a new interpretation of the cohomology group H *(ℳ g , 1 ; T [ H 1 ]) of ℳ g , 1 with coefficients in the free tensor algebra T [ H 1 ] over ℤ generated by the first homology group H 1 of Σ, by using the configuration spaces. More precisely, we define a certain cochain complex C of ℳ g , 1 ‐modules by using the configuration spaces and prove that H *(ℳ g , 1 ; C ) is canonically isomorphic to H *(ℳ g , 1 ; T [ H 1 ]).