On interrelations between graph complexes
Sergei Merkulov
TL;DR
The paper systematically analyzes Kontsevich's graph complexes $\mathsf{GC}_d$ and their oriented, sourced, and targeted relatives, introducing new interpolating complexes $\mathsf{GC}^{\mathsf{or}}_{d,d+1}$ and $\mathsf{GC}^{\mathsf t}_{d,d+1}$ that yield explicit cohomology isomorphisms with $\mathsf{GC}_d$ in adjacent dimensions. It proves a purely trivalent model $\mathsf{GC}^{\mathsf{T}}_d$ with $H^\bullet(\mathsf{GC}^{\mathsf{T}}_d)\cong H^\bullet(\mathsf{GC}_d)$, and develops a chain of reduced/directed extensions to derive two key results: (i) $H^\bullet(\mathsf{GC}_d) \cong H^\bullet(\mathsf{GC}^{\mathsf{or}}_{d+1})$ and $H^\bullet(\mathsf{GC}_d) \cong H^\bullet(\mathsf{GC}^{\mathsf t}_{d+1})$, and (ii) a short exact sequence 0 → $H^{\bullet-1}(\mathsf{GC}^{\mathsf{s\cdot t}}_{d+1})$ → $H^\bullet(\mathsf{GC}^{\mathsf{s\cdot t}}_{d+1})$ → $H^\bullet(\mathsf{GC}^{\mathsf t}_{d+1}) \oplus H^\bullet(\mathsf s_{d+1})$ → 0 that shows two copies of $H^\bullet(\mathsf{GC}_d)$ appear in the cohomology of the sourced-targeted complex. The work also situates several open problems, including questions about Lie-algebra structures on the maps and potential zig-zags of quasi-isomorphisms, and discusses a conjecture about valency-bounded reductions. Overall, the paper advances a cohesive framework connecting multiple graph complexes, offering concrete models and exact sequences that deepen understanding of their cohomology and deformation-theoretic implications.
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 $H(GC_d)$.