Triviality of the first cohomology of the even graph complex

Prove that the first cohomology group H^1(GC_2) of the even graph complex is trivial.

Background

The paper discusses the relationship between the topological differential form and the Pfaffian form in the odd graph complex GC_3, and contrasts this setting with the even graph complex GC_2. In the context of Kontsevich’s formality theorem, the authors note that failure of formality would imply non-vanishing of H1(GC_2).

The converse implication is not known: formality does not establish the vanishing of H1(GC_2). The paper therefore records the conjecture that this cohomology group is trivial, while also clarifying that the Pfaffian form studied in the paper acts on GC_3 and is not directly related to this conjecture.

References

Conversely, the formality theorem does not imply the vanishing, but still, $H1(GC_2)$ is conjectured to be trivial.

— The Topological form is the Pfaffian form  (2503.09558 - Balduf et al., 12 Mar 2025) in Section 2, Discussion, subsection “Immediate algebraic properties”