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The cohomology of the full directed graph complex. (English) Zbl 07210933
Summary: In his seminal paper “Formality conjecture”, M. Kontsevich introduced a graph complex $$\mathsf{GC}_{1ve}$$ 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 dfGC explicitly in terms of the cohomology of $$\mathsf{GC}_{1ve}$$. 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.
##### MSC:
 53D55 Deformation quantization, star products 53D17 Poisson manifolds; Poisson groupoids and algebroids 57Q45 Knots and links in high dimensions (PL-topology) (MSC2010) 18M60 Operads (general) 05C10 Planar graphs; geometric and topological aspects of graph theory 05E15 Combinatorial aspects of groups and algebras (MSC2010)
##### Keywords:
graph complexes; formality theorems
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##### References:
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