Morita classes in the homology of Aut(F_n) vanish after one stabilization

TL;DR

This paper proves that Morita classes in the rational homology of Aut(F_n) vanish after one stabilization, showing they do not persist in higher ranks, contrary to previous conjectures about their nontriviality.

Contribution

The paper demonstrates that a single stabilization map from Aut(F_n) to Aut(F_(n+1)) kills Morita classes, revealing their immediate disappearance after their initial appearance.

Findings

Morita classes vanish after one stabilization

Stability map kills Morita classes immediately

Contradicts conjecture of persistent nontrivial classes

Abstract

There is a series of cycles in the rational homology of the groups 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 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 from Aut(F_n) to Aut(F_(n+1)) kills the Morita classes, so that they disappear immediately after they appear.