Torsion in the mapping class group and its cohomology
Let \(M_ g\) be the mapping class group of a closed orientable surface of genus g. It is known that \(M_ g\) has a torsionfree subgroup of finite index and finite cohomological dimension; especially, in large dimensions with respect to g, the cohomology \(H^*(M_ g)\) has p-torsion iff p is a torsion prime for \(M_ g\). On the other hand it has been proved by \textit{J. L. Harer} [Ann. Math., II. Ser. 121, 215-249 (1985; Zbl 0579.57005)] that \(H^ k(M_ g; {\mathbb{Z}})\) is independent of g for \(g>3k\), so there exist ''stable cohomology groups'' \(H^ k(M):=H^ k(M_ g; {\mathbb{Z}})\), for \(g\gg k\). Using Chern classes of the canonical representation \(M_ g\to Gl_{2g}({\mathbb{Z}})\) induced by the action on the homology of the surface the following main theorem is proved: The stable cohomology group \(H^{4k}(M)\) contains an element of order the denominator of \(B_{2k}/2k\), where \(B_{2k}\) is the 2kth Bernoulli number. These orders grow very rapidly and involve eventually any prime power infinitely often. ''In order to prove the main theorem we need results concerning the Chern classes of representations of cyclic groups and also about the existence of torsion in \(M_ g\). We get the latter by studying suitable \({\mathbb{Z}}/n {\mathbb{Z}}\)-actions on the surface and observing that these actions eventually stabilize in the sense that they always exist for \(g\gg n.''\) See also the following review.
stable cohomology group of mapping class group of a surface
mapping class group of a closed orientable surface of genus g
Chern classes of representations of cyclic groups
scientific article; zbMATH DE number 4000997
scientific article