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Journal of the London Mathematical Society Advance Access originally published online on October 17, 2007
Journal of the London Mathematical Society 2007 76(2):451-466; doi:10.1112/jlms/jdm077
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© 2007 London Mathematical Society

The mapping class group action on the homology of the configuration spaces of surfaces

Tetsuhiro Moriyama

The Graduate School of Mathematical Sciences
The University of Tokyo
Komaba Meguro-ku
Tokyo 153
Japan

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 {Sigma} of genus g with the boundary {partial}{Sigma} {cong} S1. 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 {Sigma}. 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 Z generated by the first homology group H1 of {Sigma}, 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]).

2000 Mathematics Subject Classification 55R80 (primary). 57M05, 20F28 (secondary).

Received October 18, 2006; revised April 3, 2007; published online October 17, 2007.


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