# Homological stability for moduli spaces of manifolds

# Homological stability for moduli spaces of manifolds

THIS IS A JOINT TOPOLOGY / ALGEBRAIC TOPOLOGY SEMINAR. There will be two separate talks: 3:00-4:00 pm (Fine 214) and 4:30-5:30 pm (Fine 314). For a compact manifold $W$, possibly with boundary, we shall let $\mathrm{Diff}(W)$ denote the topological group of diffeomorphisms of $W$ fixing a neighborhood of $\partial W$. My two talks shall discuss recent joint work with Randal-Williams on the topology of the classifying space $B\mathrm{Diff}(W)$ and related spaces. An inclusion $W \subset W'$ then induces a map $B\mathrm{Diff}(W) \to B\mathrm{Diff}(W')$, and we approach the topology of $B\mathrm{Diff}(W)$ by relating it to $B\mathrm{Diff}(W')$ for other manifolds $W'$. In the first talk, I shall explain how to use infinite loop spaces to completely understand the "limiting" homology of $B\mathrm{Diff}(W)$, where the limit is over a certain direct system $W = W_0 \subset W_1 \subset \dots$, where each $W_i$ is obtained from the previous by attaching a handle. The result we prove is a higher-dimensional generalization of a theorem of Madsen and Weiss. In the second talk, I shall explain a higher-dimensional generalization of the "homological stability" theorem of J. Harer: If $W \subset W'$ is an inclusion of compact manifolds of dimension $2n$, and $W'$ is obtained from $W$ by attaching $k$-handles for $k \geq n$, then the induced map $H_*(B\mathrm{Diff}(W)) \to H_*(B\mathrm{Diff}(W'))$ is an isomorphism in a range of degrees (actually this isn't quite true, but I'll explain what the right statement is and why it's true).