Morita classes in the homology of $\mathrm{Aut}(F_n)$ vanish after one stabilization

Abstract

There is a series of cycles in the rational homology of the groups $\mathrm{Out}(F_n)$, first discovered by S. Morita, which have an elementary description in terms of finite graphs. The first two of these give nontrivial homology classes, and it is conjectured that they are all nontrivial. These cycles have natural lifts to the homology of $\mathrm{Aut}(F_n)$, which is stably trivial by a recent result of Galatius. We show that in fact a single application of the stabilization map $\mathrm{Aut}(F_n)\to \mathrm{Aut}(F_{n+1})$ kills the Morita classes, so that they disappear immediately after they appear.