Stable rational cohomology of automorphism groups of free groups and the integral cohomology of moduli spaces of graphs

Abstract

It is not known whether or not the stable rational cohomology groups $\tilde H^*(Aut(F_\infty);\Q)$ always vanish. We show that either the rational cohomology does not vanish in certain dimensions, or the integral cohomology of a moduli space of pointed graphs does not stabilize in certain other dimensions. Similar results are stated for groups of outer automorphisms. This yields that $H^5(\hat Q_m; \mathbb{Z})$, $H^6(\hat Q_m; \mathbb{Z})$, and $H^5(Q_m; \mathbb{Z})$ never stabilize as $m \to \infty$, where the moduli spaces $\hat Q_m$ and $Q_m$ are the quotients of the spines $\hat X_m$ and $X_m$ of ``outer space'' and ``auter space'', respectively, introduced by Culler and Vogtmann and by Hatcher and Vogtmann.