The topology of Ag and Its Compactifications

We survey old and new results about the cohomology of the moduli space A_g of principally polarized abelian varieties of genus g and its compactifications. The main emphasis lies on the computation of the cohomology for small genus and on stabilization results. We review both geometric and representation theoretic approaches to the problem. The appendix provides a detailed discussion of computational methods based on trace formulae and automorphic representations, in particu- lar Arthur’s endoscopic classification of automorphic representations for symplectic groups.