# Hatcher

**Cascade Topology Seminar**

Prof. Allen Hatcher

310 Malott Hall

University of Cornell

Ithaca, NY 14853 USA

http://math.cornell.edu/~hatcher

**Title: Homology of Moduli Spaces of Graphs****Abstract:** This talk will describe what is known about an analog for finite graphs of the Riemann moduli space of hyperbolic structures on a surface. The main focus to date has been on computations of the rational homology of this moduli space of graphs, which is the same as the rational homology of the automorphism group of a free group. This homology is known to stabilize as the rank of the free group increases, and Galatius showed that the stable rational homology is trivial. This is not true unstably, however, as several rational homology groups outside the stable range are known to be nontrivial. As with the Riemann moduli space, it seems to be difficult and intriguing problem to get a good picture of the unstable homology. There are general methods for constructing cycles, but the patterns for when these bound are still obscure.