On genus one mapping class groups, function spaces, and modular forms (Q2765023)

This paper gives calculations for the rational cohomology for two choices of the mapping class group. The first is formed from diffeomorphisms of a genus one surface which fix one point and leave a set of \(k\) other points invariant. If the mapping class thus formed is called \(\Gamma\) and \(\mathbb{F}\) is a field, \(\mathbb{F}\) can be considered as a \(\Gamma\)-module by noting that \(\Gamma\) maps onto the symmetric group on \(k\) elements and \(\mathbb{F}\) inherits the structure of a \(\Gamma\)-module via the sign of the permutation thus obtained. The second choice of mapping class groups uses coefficients in this sign representation and diffeomorphisms which fix a set of \(k\) points. The results are given as a bigraded group in terms of modular cusp forms based on the \(\text{SL}(2,\mathbb{Z})\) action on the upper half plane.NEWLINENEWLINEFor the entire collection see [Zbl 0972.00054].