Lukas Brantner: Outer functors, twisted arrows, and Bernoulli numbers

Abstract: The stable cohomology of \(\mathrm{Aut}(F_n)\) with polynomial coefficients is governed by Ext-groups in a category of functors on free groups. Rationally, this result of Djament was used by Vespa to recast and extend computations by Randal-Williams. More recently, Arone expressed these Ext-groups in terms of mapping spectra between functors on pointed spaces, which allowed him to obtain new integral results. For outer functors relevant to \(\mathrm{Out}(F_n)\), however, this construction no longer computes the desired Ext-groups.

I will explain how twisted arrow categories can be used to analyse this defect, and why a pattern of Bernoulli numbers emerges in the resulting calculation. This is joint work with Arone.