Algebraic & Geometric Topology

On a theorem of Kontsevich

James Conant and Karen Vogtmann

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In [‘Formal (non)commutative symplectic geometry’, The Gelfand Mathematical Seminars (1990–1992) 173–187, and ‘Feynman diagrams and low-dimensional topology’, First European Congress of Mathematics, Vol. II Paris (1992) 97–121] M Kontsevich introduced graph homology as a tool to compute the homology of three infinite dimensional Lie algebras, associated to the three operads ‘commutative,’ ‘associative’ and ‘Lie.’ We generalize his theorem to all cyclic operads, in the process giving a more careful treatment of the construction than in Kontsevich’s original papers. We also give a more explicit treatment of the isomorphisms of graph homologies with the homology of moduli space and Out(Fr) outlined by Kontsevich. In [‘Infinitesimal operations on chain complexes of graphs’, Mathematische Annalen, 327 (2003) 545–573] we defined a Lie bracket and cobracket on the commutative graph complex, which was extended in [James Conant, ‘Fusion and fission in graph complexes’, Pac. J. 209 (2003), 219–230] to the case of all cyclic operads. These operations form a Lie bi-algebra on a natural subcomplex. We show that in the associative and Lie cases the subcomplex on which the bi-algebra structure exists carries all of the homology, and we explain why the subcomplex in the commutative case does not.

Article information

Algebr. Geom. Topol., Volume 3, Number 2 (2003), 1167-1224.

Received: 5 February 2003
Revised: 1 December 2003
Accepted: 11 December 2003
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 18D50: Operads [See also 55P48]
Secondary: 57M27: Invariants of knots and 3-manifolds 32D15: Continuation of analytic objects 17B65: Infinite-dimensional Lie (super)algebras [See also 22E65]

cyclic operads graph complexes moduli space outer space


Conant, James; Vogtmann, Karen. On a theorem of Kontsevich. Algebr. Geom. Topol. 3 (2003), no. 2, 1167--1224. doi:10.2140/agt.2003.3.1167.

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