A class function on the mapping class group of an orientable surface and the Meyer cocycle

Sato, Masatoshi

doi:10.2140/agt.2008.8.1647

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A class function on the mapping class group of an orientable surface and the Meyer cocycle

Masatoshi Sato

Algebraic & Geometric Topology 8 (2008) 1647–1665

DOI: 10.2140/agt.2008.8.1647

Abstract

In this paper we define a ${Q P}^{1}$ –valued class function on the mapping class group $ℳ_{g, 2}$ of a surface $Σ_{g, 2}$ of genus $g$ with two boundary components. Let $E$ be a $Σ_{g, 2}$ –bundle over a pair of pants $P$ . Gluing to $E$ the product of an annulus and $P$ along the boundaries of each fiber, we obtain a closed surface bundle over $P$ . We have another closed surface bundle by gluing to $E$ the product of $P$ and two disks.

The sign of our class function cobounds the 2–cocycle on $ℳ_{g, 2}$ defined by the difference of the signature of these two surface bundles over $P$ .

Keywords

mapping class group, Meyer cocycle, signature cocycle

Mathematical Subject Classification 2000

Primary: 57N13, 55R40

Secondary: 57M07

References

Publication

Received: 20 February 2008

Revised: 30 May 2008

Accepted: 2 June 2008

Published: 8 October 2008

Authors

Masatoshi Sato
Graduate School of Mathematical Sciences The University of Tokyo 3-8-1 Komaba Meguro-ku Tokyo 153-8914 Japan

Volume 8, issue 3 (2008)