On the wheeled PROP of stable cohomology of Aut(F_n) with bivariant coefficients

TL;DR

This paper constructs and relates wheeled PROPs describing the stable cohomology of automorphism groups of free groups with bivariant coefficients, revealing new algebraic structures and morphisms.

Contribution

It introduces a new wheeled PROP E based on Ext-groups and constructs a morphism to the existing PROP H, connecting cohomology classes and algebraic structures.

Findings

Defined a wheeled PROP H for stable cohomology with bivariant coefficients

Constructed a wheeled PROP E from Ext-groups in functor categories

Established a morphism from E to H linking cohomology classes

Abstract

We show that the stable cohomology of automorphism groups of free groups with coefficients obtained by applying Hom(-,-) to tensor powers of the abelianization, is equipped with the structure of a wheeled PROP H. We define another wheeled PROP E by Ext-groups in the category of functors from the category of finitely generated free groups to k-modules. The main result of this paper is the construction of a morphism of wheeled PROPs $\phi: E \to H$ such that $\phi(E)$ is the wheeled PROP generated by the cohomology class h_1 constructed by the first author.