nLab
graph homology

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

Graph homology $H{𝒢}_{••}$ is the cohomology of the graph complex $({𝒢}_{••},\partial )$ which is the free $C$-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

$$\partial (\Gamma ):=\sum _{e\in E(\Gamma )\backslash \mathrm{Loop}(\Gamma )}\Gamma /e$$\partial(\Gamma) := \sum_{e\in E(\Gamma)\backslash Loop(\Gamma)} \Gamma/e

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 ${𝒢}_{\mathrm{ij}}$ is generated by those graphs which have $i$ vertices and $j$ edges; the differential has bidegree $(-1,-1)$; each ${𝒢}_{\mathrm{ij}}$ 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.

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Graph cohomology

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Applications

…moduli spaces

…deformation theory

…Rozansky-Witten theory

…Vassiliev invariants

…description of the classifying space $\mathrm{BOut}(F_n)$ of the group of outer automorphisms of a free group with $n$ generators

Graph complex controls the universal $L_{\mathrm{\infty }}$-deformations of the space of polyvector fields.

Generalizations

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

Literature

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:

• Jian Qiu, Maxim Zabzine, Introduction to graded geometry, Batalin-Vilkovisky formalism and their applications, arxiv/1105.2680
• Jian Qiu, Maxim Zabzine, Knot weight systems from graded symplectic geometry, arxiv/1110.5234