Commutative hairy graphs and representations of $Out(F_r)$

TL;DR

This paper links hairy graph complexes to representations of outer automorphism groups of free groups, providing new tools to analyze their structure and connections to deformation theory of operads.

Contribution

It introduces a novel interpretation of hairy graph complexes via decorated graph complexes associated with outer automorphism group representations.

Findings

Spectral sequence sheds light on hairy graph cohomology structure

Connection established between hairy graphs and deformation theory of little discs operads

Provides a new framework for understanding rational homotopy groups of embedding spaces

Abstract

We express the hairy graph complexes computing the rational homotopy groups of long embeddings (modulo immersion) of R^m in R^n as "decorated" graph complexes associated to certain representations of the outer automorphism groups of free groups. This interpretation gives rise to a natural spectral sequence, which allows us to shed some light on the structure of the hairy graph cohomology. We also explain briefly the connection to the deformation theory of the little discs operads and some conclusions that this brings.