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

Kawazumi, Nariya; Vespa, Christine

doi:10.2140/agt.2023.23.3089

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On the wheeled PROP of stable cohomology of $\mathrm{Aut}(F_{n})$ with bivariant coefficients

Nariya Kawazumi and Christine Vespa

Algebraic & Geometric Topology 23 (2023) 3089–3128

DOI: 10.2140/agt.2023.23.3089

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 $φ : ℰ \to ℋ$ such that $φ (ℰ)$ is the wheeled PROP generated by the cohomology class $h_{1}$ constructed by the first author.

Keywords

wheeled prop, cohomology of automorphisms of free groups

Mathematical Subject Classification

Primary: 20F28

Secondary: 18M85, 20J06

References

Publication

Received: 4 June 2021

Revised: 18 April 2022

Accepted: 3 May 2022

Published: 26 September 2023

Authors

Nariya Kawazumi
Department of Mathematical Sciences University of Tokyo Tokyo Japan

Christine Vespa
Institut de Recherche Mathématique Avancée Université de Strasbourg Strasbourg France

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Volume 23, issue 7 (2023)