Alexander Berglund: Homological stability of diffeomorphism groups

Harer's stability theorem says that the homology of the group of diffeomorphisms of an oriented genus g surface is independent of g in dimensions less than (2g-2)/3. The rational homology in this range of dimensions is known by Mumford's conjecture, proved by Madsen-Weiss.

In this talk we consider higher dimensional analogues of oriented genus g surfaces: g-fold connected sums of S^d \times S^d. In joint work with Ib Madsen we have obtained a stability theorem for the rational homology of the diffeomorphism groups of these manifolds. The proof uses rational homotopy theory and surgery theory to reduce to classical stability theorems for the homology of orthogonal and symplectic groups with twisted coefficients.