Differentials on graph complexes

Abstract

We study the cohomology of complexes of ordinary (non-decorated) graphs, introduced by M. Kontsevich. We construct spectral sequences converging to zero whose first page contains the graph cohomology. In particular, these series may be used to show the existence of an infinite series of previously unknown and provably non-trivial cohomology classes, and put constraints on the structure of the graph cohomology as a whole.

The Cohomology of the Full Directed Graph Complex

“…Using this isomorphism, we easily obtain the following generalizations of statements of Theorems 3.1 and 3.2: Theorem 3.7 Let d be any even integer and v 4m+1−d be a symbol of degree 4m + 1 − d. The natural embeddings [2], [6], [8], [16], [17], [27], [28], [29] for more details about these families of graph complexes and their generalizations. For odd d, the directions on edges play a special role.…”

“…Graph complexes provide us with a large supply of intriguing questions and conjectures 1 [3], [5], [6], [8], [9], [13], [14], [16], [17], [19], [20], [26], [27], [28], [29]. One source of the motivation for working with graph complexes comes from the study of embedding spaces [2], [4], [21], [25], [26].…”

Open problems in the Kontsevich graph construction of Poisson bracket symmetries

“…Problem 14. For a given affine Poisson manifold (M r aff , P), how many universal symmetriestaken from the countable set O r(γ ∈ ker d)(P) produced by the graph orientation morphism O r from the grt-related wheel cocycles and their iterated commutators [14,29] -do not vanish identically as global sections from Γ 2 T M r aff ? Problem 15.…”

“…In the papers [21] (see also [22] and [4,8,28] for a pedagogical review), Kontsevich introduced the graph complex -one of the many -with parity-even vertices, with a wedge ordering of parity-odd edges, and the differential d = [•−•, ·] produced by the graded commutator of graph insertions into vertices. This direction of research was furthered by Willwacher et al [11,14,29]: in particular, in [30] a generating function counts the numbers of nonzero (i.e. not equal to minus itself) unoriented graphs with respect to their bi-grading by the vertex-edge numbers.…”

Infinitesimal deformations of Poisson bi-vectors using the Kontsevich graph calculus

“…The definition of insertion γ 1 • i γ 2 of the entire graph γ 1 into vertices of γ 2 and the construction of Lie bracket [·, ·] of graphs and differential d in the non-oriented graph complex, referring to a sign convention, are as follows (cf. [8] and [7,11,12]); these definitions apply to sums of graphs by linearity. Definition 1.…”