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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.
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