Notes on Algebraic Operads, Graph Complexes, and Willwacher's Construction

TL;DR

This paper provides a detailed proof of Willwacher's theorem connecting the cohomology of the full graph complex to the deformation complex of the Gerstenhaber operad, including prerequisites and detailed constructions.

Contribution

It offers a comprehensive exposition and proof of Willwacher's theorem, clarifying the relationship between graph complexes and operad deformation theory.

Findings

Proof of Willwacher's theorem established

Detailed exposition of operads, cooperads, and the cobar construction

Enhanced understanding of the convolution Lie algebra and twisting construction

Abstract

We give a detailed proof of T. Willwacher's theorem arXiv:1009.1654 which links the cohomology of the full graph complex fGC to the cohomology of the deformation complex of the operad GER, governing Gerstenhaber algebras. We also present various prerequisites required for understanding the material of arXiv:1009.1654. In particular, we review operads, cooperads, and the cobar construction. We give a detailed exposition of the convolution Lie algebra and its properties. We prove a useful lifting property for maps from a dg operad obtained via the cobar construction. We describe in detail Willwacher's twisting construction, and then use it to work with various operads assembled from graphs, in particular, the full graph complex and its subcomplexes. These notes are loosely based on lectures given by the first author at the Graduate and Postdoc Summer School at the Center for Mathematics at Notre Dame (May 31 - June 4, 2011).