Differentials on graph complexes II - hairy graphs

TL;DR

This paper investigates the cohomology of hairy graph complexes related to embedding spaces, introducing spectral sequences and a novel waterfall mechanism to construct new cohomology classes, revealing insights into their global structure.

Contribution

It introduces spectral sequences converging to zero and the waterfall mechanism to generate new hairy graph cohomology classes from non-hairy classes.

Findings

Spectral sequences with initial pages containing hairy graph cohomology.

A new waterfall mechanism for constructing cohomology classes.

First insights into the global structure of hairy graph cohomology.

Abstract

We study the cohomology of the hairy graph complexes which compute the rational homotopy of embedding spaces, generalizing the Vassiliev invariants of knot theory. We provide spectral sequences converging to zero whose first pages contain the hairy graph cohomology. Our results yield a way to construct many hairy graph cohomology classes out of non-hairy classes by a mechanism which we call the waterfall mechanism. By this mechanism we can construct many previously unknown classes and provide a first glimpse at the tentative global structure of the hairy graph cohomology.