For any integer d, the Kontsevich graph complex GC^2_d and the oriented graph complex OGC^2_{d+1} are connected by a zigzag of quasi-isomorphisms of dg Lie algebras.
On interrelations between graph complexes
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abstract
We study Maxim Kontsevich's graph complex $GC_d$ for any integer $d$ as well as its oriented and targeted versions, and show new short proofs of the theorems due to Thomas Willwacher and Marko Zivkovic which establish isomorphisms of their cohomology groups. A new result relating the cohomology of the sourced-targeted graph complex in dimension $d+1$ with the direct sum of two copies of the cohomology group of Maxim Kontsevich's graph complex $GC_d$ in dimension $d$ is obtained. We introduce a new graph complex spanned by purely trivalent graphs and show that its cohomology is isomorphic to $H(GC_d)$.
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The oriented graph complex revisited
For any integer d, the Kontsevich graph complex GC^2_d and the oriented graph complex OGC^2_{d+1} are connected by a zigzag of quasi-isomorphisms of dg Lie algebras.