Graph complexes and their cohomology

Abstract

In this thesis we study cohomologies of two kind of graph complexes, Kontsevich graph complexes and hairy graph complexes, by three dfferent methods. In the first method we combinatorially calculate the dimensions of Kontsevich graph complexes by providing generating functions, leading to the Euler characteristics. Secondly, we prove that multiple edges may be omitted without essentially changing the cohomology in both type of complexes. The third method introduces extra dfferentials on all complexes, leading to spectral sequences that converge to (almost) zero with the standard dfferential being the first one. This leads to the conclusion that classes of standard cohomology come in pairs.