Fusion and Fission in Graph Complexes

We analyze a functor from cyclic operads to chain complexes first considered by Getzler and Kapranov and also by Markl. This functor is a generalization of the graph homology considered by Kontsevich, which was defined for the three operads Comm, Assoc, and Lie. More specifically we show that these chain complexes have a rich algebraic structure in the form of families of operations defined by fusion and fission. These operations fit together to form uncountably many Lie∞ and co-Lie∞ structures. In particular, the chain complexes have a bracket and cobracket which are compatible in the Lie bialgebra sense on a certain natural subcomplex.