Topology Seminar: Soren Galatius (Stanford)

We study the space M_g of isometry classes (or conformal equivalence classes) of smooth manifolds, diffeomorphic to #^g(S^d \times S^d), the connected sum of g copies of S^d \times S^d. For 2d=2, this is essentially the moduli space of Riemann surfaces. There is a variant M_{g,1} where we consider moduli of manifolds with an embedded D^{2d}; connected sum with S^d \times S^d gives a map M_{g,1} \to M_{g+1,1}, and we can form the direct limit M_{\infty,1}. The work of Madsen and Weiss on Mumford's conjecture determines the homology of M_{\infty,1} in the case 2d=2. We give a similar description of the homology of M_{\infty,1} in higher dimensions (2d \geq 6). This is joint work with Oscar Randal-Williams.