The cohomology of the full directed graph complex

TL;DR

This paper explicitly describes the cohomology of the full directed graph complex using the cohomology of a related graph complex, with applications to classifying formality quasi-isomorphisms for Hochschild cochains.

Contribution

It provides an explicit cohomology description of the full directed graph complex, linking it to existing complexes and applications in Hochschild cohomology.

Findings

Explicit cohomology formula for the full directed graph complex

Connection established between the full directed graph complex and $GC_{1ve}$

Applications to classifying formality quasi-isomorphisms in Hochschild cohomology

Abstract

In his seminal paper "Formality conjecture", M. Kontsevich introduced a graph complex $GC_{1ve}$ closely connected with the problem of constructing a formality quasi-isomorphism for Hochschild cochains. In this paper, we express the cohomology of the full directed graph complex explicitly in terms of the cohomology of $GC_{1ve}$. Applications of our results include a recent work by the first author which completely characterizes homotopy classes of formality quasi-isomorphisms for Hochschild cochains in the stable setting.