Stable cohomology of the universal Picard varieties and the extended mapping class group (Q455495)

The paper under review studies different aspects concerned with the cohomology of universal Picard varieties and of extended mapping class groups.NEWLINENEWLINELet \(\mathcal S_{g,r}^n\) be the moduli space of smooth surfaces of genus \(g\) with \(r\) boundary components, \(n\) marked points and a complex line bundle, originally introduced by \textit{R. L. Cohen} and \textit{I. Madsen} [Proc. Symp. Pure Math. 80, Pt. 1, 43--76 (2009; Zbl 1210.57029)] and \(\mathcal S_{g,r}^n(k)\) its path component corresponding to bundles of degree \(k\). The extended mapping class group \(\widetilde{\Gamma}_{g,r}^n(k)\) is the fundamental group of the space \(\mathcal S_{g,r}^n(k)\), based at a fixed bundle, which for \((r,n)=(1,0)\) coincides with \textit{N. Kawazumi}'s definition in [Invent. Math. 131, No.1, 137--149 (1998; Zbl 0894.57020)].NEWLINENEWLINEThe authors start by stating that, for \(r+n>0\), the cohomology of the group \(\widetilde{\Gamma}_{g,r}^n(k)\) can be obtained from the cohomology of \(\mathcal S_{g,r}^n(k)\) as there is an homotopy equivalence \(B\widetilde{\Gamma}_{g,r}^n(k)\simeq \mathcal S_{g,r}^n(k)\). For computing the cohomology of \(\mathcal S_{g,r}^n(k)\), and as a consequence of its moduli-theoretic interpretation, the authors use a number of stability type results that they collect in the paper and use auxiliary infinite loop spaces for doing calculations. Special attention is needed to treat the case when \(r=n=0\), as the relation between the cohomologies of \(\mathcal S_g(k):=\mathcal S_{g,0}^0(k)\) and \(\widetilde \Gamma_g(k):=\widetilde{\Gamma}_{g,0}^0(k)\) is more subtle; the authors use the existence of a fibre sequence relating both to obtain results for the cohomology of \(B\widetilde{\Gamma}_{g}(k)\) from that of \(\mathcal S_g(k)\).NEWLINENEWLINEIn the last part of the paper the authors explain how their results translate to questions on complex algebraic geometry. Let \(\text{Hol}_g(k)\) be the moduli stack classifying families of Riemann surfaces of genus \(g\) equipped with a holomorphic line bundle of degree \(k\), which is a \(\mathbb C^*\) gerbe over universal Picard stack \(\text{Pic}_g(k)\). The stacks \(\text{Hol}_g(k)\) and \(\text{Pic}_g(k)\) are shown to be homotopic equivalent to \(B\widetilde \Gamma_g(k)\) and to \(\mathcal S_g(k)\), respectively. As a consequence, the authors describe explicit generators for the holomorphic Picard groups of \(\text{Pic}_g(k)\) and of \(\text{Hol}_g(k)\), in particular recovering results by \textit{A. Kouvidakis} in [J. Differ. Geom. 34, No. 3, 839--850 (1991; Zbl 0780.14004)] for \(\text{Pic}_g(k)\).