The heptagon-wheel cocycle in the Kontsevich graph complex

Abstract

The real vector space of non-oriented graphs is known to carry a differential graded Lie algebra structure. Cocycles in the Kontsevich graph complex, expressed using formal sums of graphs on n vertices and 2n — 2 edges, induce - under the orientation mapping - infinitesimal symmetries of classical Poisson structures on arbitrary finite-dimensional affine real manifolds. Willwacher has stated the existence of a nontrivial cocycle that contains the (2ℓ+1)-wheel graph with a nonzero coefficient at every ℓ∈ℕ. We present detailed calculations of the differential of graphs; for the tetrahedron and pentagon-wheel cocycles, consisting at ℓ = 1 and ℓ = 2 of one and two graphs respectively, the cocycle condition d(γ) = 0 is verified by hand. For the next, heptagon-wheel cocycle (known to exist at ℓ = 3), we provide an explicit representative: it consists of 46 graphs on 8 vertices and 14 edges.