Commutative hairy graphs and representations of Out(F-r)

摘要

We express the hairy graph complexes computing the rational homotopy groups of long embeddings (modulo immersions) 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 disks operads and some conclusions that this brings.