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13 Aug 2003
Infinitesimal operations on complexes of graphs
 Jim Conant,
 Karen Vogtmann
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In two seminal papers Kontsevich used a construction called graph homology as a bridge between certain infinite dimensional Lie algebras and various topological objects, including moduli spaces of curves, the group of outer automorphisms of a free group, and invariants of odd dimensional manifolds. In this paper, we show that Kontsevich’s graph complexes, which include graph complexes studied earlier by Culler and Vogtmann and by Penner, have a rich algebraic structure. We define a Lie bracket and cobracket on graph complexes, and in fact show that they are BatalinVilkovisky algebras, and therefore Gerstenhaber algebras. We also find natural subcomplexes on which the bracket and cobracket are compatible as a Lie bialgebra. Kontsevich’s graph complex construction was generalized to the context of operads by Ginzburg and Kapranov, with later generalizations by GetzlerKapranov and Markl. In [CoV], we show that Kontsevich’s results in fact extend to general cyclic operads. For some operads, including the examples associated to moduli space and outer automorphism groups of free groups, the subcomplex on which we have a Lie bialgebra structure is quasiisomorphic to the entire connected graph complex. In the present paper we show that all of the new algebraic operations canonically vanish when the homology functor is applied, and we expect that the resulting constraints will be useful in studying the homology of the mapping class group, finite type manifold invariants and the homology of Out(F _{ n }).
Mathematics Subject Classification (2000): 17B62, 17B63, 17B70, 20F28, 57M07, 57M15, 57M27
Partially supported by NSF VIGRE grant DMS9983660
Partially supported by NSF grant DMS9307313
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 Title
 Infinitesimal operations on complexes of graphs
 Journal

Mathematische Annalen
Volume 327, Issue 3 , pp 545573
 Cover Date
 20031101
 DOI
 10.1007/s0020800304652
 Print ISSN
 00255831
 Online ISSN
 14321807
 Publisher
 SpringerVerlag
 Additional Links
 Authors

 Jim Conant ^{(1)}
 Karen Vogtmann ^{(1)}
 Author Affiliations

 1. Department of Mathematics, Cornell University, 310 Malott Hall, Ithaca, NY 148534201, USA