This entry will be (for now) both about graph homology and about graph cohomology, which were originally introduced by Maxim Kontsevich; once the material grows, it can be separated into two entries. Kontsevich produced few version of the graph complex, the basic one attached to the operad of 3-valent ribbon graphs. A graph complex can be produced as an output from the Feynman transform of a modular operad.
Graph homology $H\mathcal{G}_{\bullet\bullet}$ is the cohomology of the graph complex $(\mathcal{G}_{\bullet\bullet}, \partial)$ which is the free $\mathbf{C}$-vector space generated by isomorphism classes of oriented ribbon graphs modulo relation $(\Gamma,-\sigma) = - (\Gamma, \sigma)$ where $\Gamma$ is a ribbon graph with orientation $\sigma$. The differential is given by
where the sum is over edges $e$ which are not loops and $\Gamma/e$ is obtained from $\Gamma$ by contraction at edge $e$ (cf. ribbon graph). The map $\partial$ is really a differential ($\partial^2 = 0$) because two contractions in different order produce a different orientation. There is a canonical bigrading on the graph complex, where $\mathcal{G}_{ij}$ is generated by those graphs which have $i$ vertices and $j$ edges; the differential has bidegree $(-1,-1)$; each $\mathcal{G}_{ij}$ is finite-dimensional, while the whole complex is infinte-dimensional. Graph splits into a direct sum of subcomplexes labelled by the Euler characteristics of the underlying graph. The structure of a graph complex reflects a structure in the Chevalley-Eilenberg complex of a certain Lie algebra; and the graph homology to the relative Lie homology of that Lie algebra as shown by Kontsevich.
…
…
The Graph complex carries the structure of a dg-Lie algebra (L-infinity algebra) which acts on the space of choices of formal deformation quantization of Poisson manifolds. Its degree-0 chain homology is the Lie algebra of the Grothendieck-Teichmüller group.
The homology in negative degree vanishes and that in positive degree is still unknown, but computer experiements show that at least the third cohomology contains nontrivial elements.
The degree-0 homology is also isomorphic, up to one “scaling class”, to the 0th cohomology of the derivations of the E2 operad.
See at deformation quantization – Motivic Galois group action on the space of quantizations.
See at Grothendieck-Teichmüller group – relation to the graph complex.
…moduli spaces
…deformation theory
…Rozansky-Witten theory
…Vassiliev invariants
…description of the classifying space $BOut(F_n)$ of the group of outer automorphisms of a free group with $n$ generators
Graph complex controls the universal $L_\infty$-deformations of the space of polyvector fields.
There are generalizations for $d$-algebras (algebras over little disc operad in higher dimension). The cohomological graph complex is then the case for $d=2$. There is also a “directed” version. On the other hand, graph complex
Cf. also Rozansky-Witten theory, formal noncommutative symplectic geometry
Maxim Kontsevich, Feynman diagrams and low-dimensional topology, First European Congress of Mathematics, 1992, Paris, vol. II, Progress in Mathematics 120, Birkhäuser (1994), 97–121, pdf
Maxim Kontsevich, Rozansky–Witten invariants via formal geometry, Compositio Mathematica 115: 115–127, 1999, doi, arXiv:dg-ga/9704009
Martin Markl, Steve Shnider, James D. Stasheff, Operads in algebra, topology and physics, Math. Surveys and Monographs 96, Amer. Math. Soc. 2002.
Andrey Lazarev, Operads and topological conformal field theories, pdf; and older versio: Graduate lectures on operads and topological field theories, zip file with 11 pdfs, over 5 Mb
Alastair Hamilton, Andrey Lazarev, Graph cohomology classes in the Batalin-Vilkovisky formalism, J.Geom.Phys. 59:555-575, 2009, arxiv/0701825
Alastair Hamilton, A super-analogue of Kontsevich’s theorem on graph homology, Lett. Math. Phys. 76 (2006), no. 1, 37–55, math.QA/0510390
A. Lazarev, A. A. Voronov, Graph homology: Koszul and Verdier duality, math.QA/0702313
M. V. Movshev, A definition of graph homology and graph K-theory of algebras, math.KT/9911111
Alberto S. Cattaneo, Paolo Cotta-Ramusino, Riccardo Longoni, Algebraic structures on graph cohomology, Journal of Knot Theory and Its Ramifications, Vol. 14, No. 5 (2005) 627-640, doi, math.GT/0307218 , MR2006g:58021)
K. Igusa, Graph cohomology and Kontsevich cycles, Topology 43 (2004), n. 6, p. 1469-1510, MR2005d:57028, doi
Thomas Willwacher, M. Kontsevich’s graph complex and the Grothendieck-Teichmueller Lie algebra, arxiv/1009.1654
Vasily Dolgushev, Christopher L. Rogers, Thomas Willwacher, Kontsevich’s graph complex, GRT, and the deformation complex of the sheaf of polyvector fields, arxiv/1211.4230
Damien Calaque, Carlo A. Rossi, Lectures on Duflo isomorphisms in Lie algebra and complex geometry, European Math. Soc. 2011
S. A. Merkulov, Graph complexes with loops and wheels, in (Manin’s Festschrift:) Algebra, Arithmetic, and Geometry, Progress in Mathematics 270 (2009) 311-354, doi, pdf
The following survey has discussion of context between the graph complex and Batalin-Vilkovisky formalism: