Kontsevich Graph Cohomology

Updated 20 August 2025

Kontsevich graph cohomology is a combinatorial dg Lie algebra framework built from trivalent graph complexes that reveals deep algebraic and topological properties.
It bridges deformation quantization, Lie algebra cohomology, and quantum topology, establishing links with the Grothendieck–Teichmüller Lie algebra and moduli space geometry.
Advanced computational techniques using spectral sequences and subcomplex decompositions uncover new cohomology classes and highlight explosive growth in higher loop orders.

Kontsevich graph cohomology refers to a family of cohomological theories built from graph complexes introduced by Maxim Kontsevich, initially as part of his work on deformation quantization. These cohomology theories are defined by associating combinatorial dg Lie algebras (or related structures) to spaces of graphs—often trivalent, sometimes with additional decorations—and investigating the algebraic, topological, and representation-theoretic properties of the resulting cohomology spaces. Kontsevich graph cohomology has profound connections to Lie algebra cohomology, deformation theory, the Grothendieck–Teichmüller Lie algebra, quantum topology, and the geometry of moduli spaces.

1. Foundations: Definitions and Constructions

At the core is the Kontsevich graph complex (commutative case), denoted GC. Its generators are (isomorphism classes of) finite, connected, trivalent graphs (typically without self-loops or multiple edges, though variants exist), equipped with an orientation (orderings of edges or, for ribbon graphs, cyclic orderings at vertices). The grading is given by formulas such as

$|\Gamma| = - (n-1) \cdot \# \text{edges} + n \cdot (\#\text{vertices} - 1)$

for, say, $\mathrm{GC}_n$ . The differential $d$ is defined by summing over all possible edge contractions: $d \Gamma = \sum_{e} (\Gamma / e, o_e)$ where contraction respects the orientation data.

Variants include:

Full graph complexes (allowing vertices of valence $\geq 3$ or even bivalent vertices)
Directed or “aerial” graph complexes (with directed edges, relevant to polyvector fields)
Decorated, hairy, and ribbon graph complexes (used in topological applications)
Bridgeless graph complexes (spanned by 2-edge-connected graphs) (Willwacher, 25 Mar 2025)

The cochain complex structure supports a dg Lie algebra structure, with the Lie bracket obtained by “insertion” of graphs at vertices.

2. Cohomology of the Graph Complex and Odd Wheels

A fundamental result is that, in many cases, the (Chevalley) cohomology of relevant Lie algebras or operads—when computed with graph-cocycle-valued cochains—is freely generated by wedge products of “odd wheels” (Aloulou et al., 2010). Specifically, in the setting of the Lie algebra of ascending tensors (ascending meaning coefficients of degree $q$ for a $p$ -tensor with $q < p$ if $p > 1$ ), the subspace cohomology generated by graph cochains is

$\bigwedge_{i} R_{2k_i+1}$

where $R_m$ denotes the $m$ -vertex wheel (an $m$ -cycle), $k_1 < k_2 < \dotsc < k_p$ , and $\sum (2k_i+1) = n$ (for cohomological degree $n$ ). Cochains associated to wheels of length $> 2d$ in $\mathbb{R}^d$ vanish.

This reliance on odd wheels is also reflected in the algebraic generators of the Grothendieck–Teichmüller Lie algebra and in the stable cohomology of polyvector fields (Willwacher, 2011).

3. Algebraic and Operadic Links: Grothendieck–Teichmüller Lie Algebra, Symmetries

Kontsevich’s graph cohomology establishes a deep link with the Grothendieck–Teichmüller Lie algebra $\mathfrak{grt}_1$ (Willwacher, 2010, Andersson et al., 2019). Specifically, the degree-zero cohomology $H^0(\mathrm{GC})$ is isomorphic to $\mathfrak{grt}_1$ , with the correspondence realized by associating to each graph cocycle the unique infinitesimal symmetry it generates on deformation and formality morphisms.

This action is canonical, and explicit algorithms exist to read off $\mathfrak{grt}_1$ -elements from graph cocycles, with the coefficients in front of odd wheel graphs determining the image in the Lie algebra. The action of $\mathfrak{grt}_1$ via graph cohomology also governs the (homotopy) automorphisms of operads such as ho $(e_2)$ and, further, the universal deformation theory of $E_n$ -operads and related objects.

4. Geometric and Topological Applications

Kontsevich graph cohomology plays a crucial role in several geometric contexts:

Deformation quantization and formality: The complex governs the analytic deformation theory of the Schouten algebra of polyvector fields, producing universal deformation classes via cocycles in the graph complex (Morand, 2019, Dolgushev et al., 2012). Universal formality morphisms for Hochschild cochains are classified by the cohomology of the complex and the action of $\mathfrak{grt}_1$ .
Moduli spaces and tropical geometry: The homology of the (commutative) graph complex computes the top weight graded cohomology of the moduli space of curves $\mathcal{M}_g$ (Chan et al., 2018). There is a canonical isomorphism, up to a degree shift, between the homology of the Kontsevich graph complex and the reduced homology of the dual complex (at infinity) of the tropical moduli space:

$\operatorname{Gr}_{6g-6}^W H^{6g-6-k}(\overline{\mathcal{M}_g};\mathbb{Q}) \cong \widetilde{H}_{k-1}(\Delta_g; \mathbb{Q})$

and this further implies exponential and even super-exponential growth of unstable cohomology in high genus (Borinsky et al., 7 May 2024).

Knot invariants and finite type theory: The graph complex (and associated “hairy” variants) provides the algebraic framework underlying the paper of Vassiliev (finite type) knot invariants, with trivalent graph cocycles yielding integer-valued cohomology classes on spaces of knots via configuration space integral constructions (Koytcheff, 2015). This is paralleled in the connection to the Kontsevich integral in quantum topology, where the graph complex organizes the algebra of chord diagrams, modulo relations.

5. Spectral Sequences, New Classes, and Structural Constraints

The graph cohomology’s structure admits spectral sequences that reveal intricate patterns, including the existence of infinite series of previously unknown, nontrivial cohomology classes, especially in degrees outside the “dominant” range (Khoroshkin et al., 2014). Spectral sequences show “pairing phenomena” where classical classes (such as grt $_1$ elements) are eventually killed by “partner” cohomology classes (e.g., loop cocycles), tightly constraining the possible structure of graph cohomology in each degree and loop order.

An important computational development is the reduction to subcomplexes such as the triconnected and bridgeless graph complexes, supported by combinatorial and operadic decomposition techniques (Willwacher, 25 Mar 2025, Willwacher, 19 Aug 2025). The cohomology of the full complex can be reconstructed from that of such subcomplexes via cyclic bar constructions and spectral sequences, with explicit quasi-isomorphisms relating the complexes.

6. Variants, Generalizations, and Representation-Theoretic Aspects

Graph cohomology has extensive generalizations, including:

Directed and cyclic graph complexes: For example, the “full directed graph complex” (dfGC) is explicitly described in terms of the cohomology of the 1-vertex-irreducible subcomplex (Dolgushev et al., 2017).
Hairy graph complexes and cyclic operads: These variants encode additional data (e.g., symplectic labelings as “hairs” or cyclic operad colorings at vertices) and relate hairy graph homology to dihedral homology of associative algebras, with implications for Out $(F_n)$ and the Johnson cokernel (Conant et al., 2013).
Geometric and cohomological formalisms: Concepts such as emeralds, relaxed Čech cohomology, and configuration space sheaves provide new categorical frameworks for interpreting graph cohomology in the context of decorated configuration spaces and the Kontsevich integral (Gauthier, 2012).
Algebraic and symplectic representations: The action of $\mathfrak{grt}_1$ via graph cocycles is reflected in explicit operations on (stable) Lie algebra cohomology, connections to characteristic classes (e.g., contraction with odd Chern character components), and S $p$ (V)-module decompositions in hairy graph theory (Dolgushev et al., 2012, Willwacher, 2011, Conant et al., 2013).

7. Computational and Asymptotic Aspects

State-of-the-art computations now reach double-digit loop orders (Willwacher, 19 Aug 2025). The dimension of the graph homology exhibits explosive growth with increasing loop order or genus, with explicit asymptotic formulas showing super-exponential expansion governed by factorial terms derived from Bernoulli numbers (Borinsky et al., 7 May 2024): $\dim H(GC_g^+) \geq (C \cdot g)^g$ for some constant $C>0$ . The growth of the homology has major implications for the topology of moduli spaces and the understanding of non-tautological cohomology classes.

Advanced algorithms and computer-assisted methods (including sparse matrix rank computations over finite fields, spectral sequence calculations, and combinatorial reductions) drive new high-loop computations and the discovery of left-out classes or cancellations, such as counterexamples to strong forms of conjectures on the structure of the Grothendieck–Teichmüller Lie algebra in graph cohomology (Willwacher, 19 Aug 2025).

Kontsevich graph cohomology thus provides a universal algebraic framework capturing the deformation, symmetry, and topological properties of a broad range of mathematical structures—polyvector fields, operads, moduli spaces, knot spaces—via a combinatorial complex whose structure, computation, and applications continue to shape developments in algebraic topology, mathematical physics, and algebraic geometry.