Cohomology of the undeformed Kontsevich graph complex

Determine the cohomology of the Kontsevich graph complex \(GC_0\) equipped with the vertex-splitting differential \(d=\frac{1}{2}[L,-]\).

Background

The paper distinguishes the undeformed graph complex (GC0,d)(GC_0,d), whose differential is induced by the Maurer–Cartan element L/2L/2, from the twisted complex (GC0,d+∇)(GC_0,d+\nabla), where ∇\nabla is induced by the tadpole graph. The twisted complex has a completely computed cohomology generated by the tadpole and odd wheel-related classes, whereas the corresponding cohomology of the undeformed complex is not determined.

Understanding H(GC0,d)H(GC_0,d) would clarify the structure of the ordinary Kontsevich graph complex and its relationship to the more tractable twisted complex used to model the rational K-theory of Z\mathbb Z.

References

While the cohomology of (GC_0,d) is largely unknown, the cohomology of the twisted complex (GC_0,d+\nabla) can be fully computed:

— The graph complex and the homology of the general linear groups revisited  (2610.12263 - Patzt et al., 8 Oct 2026) in Section 3, subsection “Deformed differential \(\nabla\)”