A generalization of the Morita-Mumford classes to extended mapping class groups for surfaces
Let \(\Sigma^n_{g,r}\) be the 2-manifold with genus \(g\), \(r\) boundary circles, and \(n\) punctures. Denote the mapping class group, fixing the boundary and punctures, of \(\Sigma^n_{g,r}\) by \(\Gamma^n_{g,r}\). Let \(\Sigma_{g,r}=\Sigma^0_{g,r}\) and \(\Sigma_g=\Sigma^0_{g,0}\). \(\Gamma^n_{g,r}\) acts on \(\Sigma_{g,1}\) and thus on \(H_1(\Sigma_{g,1})\). Define \(\widetilde{\Gamma}^n_{g,r}\) to be the semi-direct product \(H_1(\Sigma_{g,1})\circ\Gamma^n_{g,r}\), the extended mapping class group. This paper defines bigraded series of cohomology classes, \(m_{i,j}\) and \(\widetilde{m}_{i,j}\) in \(H^{2i+j-2}(\Gamma_{g,1};\wedge^j H)\) and \(H^{2i+2j-2}(\widetilde{\Gamma}_{g,1},\mathbb{Z})\) respectively, where \(H = H_1(\Sigma_{g,1}; \mathbb{Z})\). \(m_{i+1,0}\) and \(\widetilde{m}_{i+1,0}\) are equal to the \(i\)th Morita-Mumford class \(e_i\) in \(H^{2i}(\Gamma_{g,1};\mathbb{Z})\). The classes \(m_{2,0}\), \(m_{1,1}\) and \(m_{0,3}\) are explicitly calculated. It is shown for \(r+s=2\) and the case \(g\geq 5\) with \((r,s)= (1,3)\) that the \(m_{i,j}\)'s generate \(H^r(\Gamma_{g,1};\wedge^s H_1(\Sigma_{g,1};\mathbb{Q}))\).
scientific article; zbMATH DE number 1146051