Differentials on graph complexes

TL;DR

This paper investigates the cohomology of graph complexes introduced by Kontsevich, using spectral sequences to identify new non-trivial classes and analyze the structure of the cohomology.

Contribution

It constructs spectral sequences converging to zero with initial pages revealing new non-trivial graph cohomology classes and constraints on their structure.

Findings

Existence of an infinite series of non-trivial cohomology classes.

Spectral sequences provide a method to analyze graph cohomology.

Constraints on the overall structure of graph cohomology.

Abstract

We study the cohomology of complexes of ordinary (non-decorated) graphs, introduced by M. Kontsevich. We construct spectral sequences converging to zero whose first page contains the graph cohomology. In particular, these series may be used to show the existence of an infinite series of previously unknown and provably non-trivial cohomology classes, and put constraints on the structure of the graph cohomology as a whole.