Differentials on Graph Complexes III - Deleting a Vertex

TL;DR

This paper introduces a new differential called deleting a vertex in hairy graph complexes, demonstrating its near-acyclicity and enabling the construction of cohomology classes via the waterfall mechanism.

Contribution

It presents a novel differential in hairy graph complexes and proves its near-acyclicity, facilitating new methods for constructing graph cohomology classes.

Findings

Hairy graph complex with the new differential is almost acyclic

The waterfall mechanism for cohomology class construction is justified

Potential for further applications of the deleting a vertex differential

Abstract

We prove that the hairy graph complex with the extra differential is almost acyclic. This justifies the construction of hairy graph cohomology classes by the waterfall mechanism. The main idea used in the paper is a new differential, deleting a vertex. We hope that the new differential can have further applications in the study of graph cohomology.