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On a theorem of Kontsevich. (English) Zbl 1063.18007

Operads were invented in algebraic topology to encapsulate the operations of (that is, to provide the abstract theory for) homotopy invariant algebraic structures. The operads in the three special cases of commutative, associative and Lie algebras have the further property of being cyclic.
This paper reviews cyclic operads using a calculus of spider-like diagrams. The more general Getzler-Kapranov chain complex [E. Getzler and M. M. Kapranov, “Modular operads”, Compos. Math. 110, 65–126 (1998; Zbl 0894.18005)] is described here to yield a functor from cyclic operads to symplectic Lie algebras. The values of this functor on the three special cyclic operads mentioned above are infinite-dimensional Lie algebras whose homologies were computed by M. Kontsevich in [“Formal (non)commutative symplectic geometry”, The Gelfand Seminars, 1990–1992, 173–187 (1993; Zbl 0821.58018) and “Feynman diagrams and low-dimensional topology”, Prog. Math. 120, 97–121 (1994; Zbl 0872.57001)].
In developing the present careful account of Kontsevich’s work, the authors discovered a gap in the proof of the main theorem relating symplectic invariants and graph homology. Kontsevich suggested a correction in the commutative case which the authors have adapted to a general cyclic operad. They also provide an explicit treatment of the identification, only outlined by Kontsevich, of graph homologies in terms of moduli and outer space homologies.

MSC:

18D50 Operads (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
32D15 Continuation of analytic objects in several complex variables
17B65 Infinite-dimensional Lie (super)algebras
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