Spaces of graphs and surfaces: on the work of Søren Galatius

Abstract

Galatius’ most striking result is easy enough to state. Let Σn be the symmetric group on n letters and Fn be the free (non-abelian) group with n generators. The symmetric group Σn acts naturally by permutation on the n generators of Fn, and every permutation gives thus rise to an automorphism of the free group. Galatius proves that the map Σn → AutFn in homology induces an isomorphism in degrees less than (n− 1)/2. The homology of the symmetric groups in these ranges is well understood. In particular, in common with all finite groups, it has no non-trivial rational homology. By Galatius’ theorem, in low degrees this then is also true for AutFn: H∗(AutFn)⊗Q = 0 for 0 < ∗ < 2n/3,