Infinitesimal operations on complexes of graphs
 Jim Conant,
 Karen Vogtmann
 … show all 2 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract.
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 }).
 BarNatan, D., Garoufalidis, S., Rozansky, L., Thurston, D.: The Aarhus integral of rational homology 3spheres I: A highly nontrivial flat connection on S ^{3}. To appear in Selecta Mathematica
 Conant, J.: Fusion and fission in graph complexes. Pacific J. Math. 209, 219–230 (2003)
 Conant, J., Vogtmann, K.: On a theorem of Kontsevich. math.QA/0208169
 Conant, J., Gerlits, F., Vogtmann, K.: Cut vertices in commutative graphs. Preprint.
 Culler, M., Vogtmann, K.: Moduli of graphs and automorphisms of free groups. Invent. Math. 84, 91–119 (1986)
 Chas, M., Sullivan, D.: String topology. To appear in Annals of Math, math.GT/ 9911159
 Chas, M.: Combinatorial Lie bialgebras of curves on surfaces. To appear in Topology, math.GT/0105178
 Garoufalidis, S., Levine, J.: Tree level invariants of threemanifolds, massey products and the Johnson homomorphism. Preprint 1999, math.GT/9904106
 Getzler, E., Kapranov, M.M.: Cyclic operads and cyclic homology. Geometry, topology, and physics, Conf. Proc. Lecture Notes Geom. Topology, IV, Internat. Press, Cambridge, MA, 1995, pp. 167–201
 Ginzburg, V., Kapranov, M.: Koszul duality for operads. Duke Math. J. 76, 203–272 (1994)
 Getzler, E., Kapranov, M.M.: Modular operads. Compositio Math. 110, 65–126 (1998) CrossRef
 Gerlits, F.: Calculations in graph homology. In preparation
 Getzler, E.: BatalinVilkovisky algebras and twodimensional topological field theories. Preprint 2001
 Kontsevich, M.: Formal (non)commutative symplectic geometry. The Gelfand Mathematical Seminars, 1990–1992, Birkhüser Boston, Boston, MA, 1993, pp. 173–187
 Kontsevich, M.: Feynman diagrams and lowdimensional topology. First European Congress of Mathematics, Vol. II (Paris, 1992), Progr. Math., 120, Birkhüser, Basel, 1994, pp. 97–121
 Kuperberg, G., Thurston, D.P.: Perturbative 3manifold invariants by cutandpaste topology. Preprint 1999, UC Davis Math 199936, math.GT/9912167
 Le, T.Q.T., Murakami, J., Ohtsuki, T.: On a universal perturbative invariant of 3manifolds. Topology 373 (1998)
 Levine, J.: Homology cylinders, an expansion of the mapping class group. Algebr. Geom. Topol. 1, 551–578 (2001)
 Majid, S.: Foundations of quantum group theory. Cambridge University Press, Cambridge, 1995, pp. x+607
 Markl, M.: Cyclic operads and homology of graph complexes. Rendiconti del circolo matematico di Palermo, serie II, Suppl. 59, 161–170 (1999)
 Milnor, J.W., Moore, J.C.: On the structure of Hopf algebras. Ann. Math. 81(2), 211–264 (1965)
 Penner, R.C.: Perturbative series and the moduli space of Riemann surfaces. J. Differ. Geom. 27, 35–53 (1988)
 Thurston, D.: Undergraduate thesis, Harvard
 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