Cyclic operads and homology of graph complexes

We will consider P-graph complexes, where P is a cyclic operad. P-graph complexes are natural generalizations of Kontsevich's graph complexes -- for P = the operad for associative algebras it is the complex of ribbon graphs, for P = the operad for commutative associative algebras, the complex of all graphs. We construct a `universal class' in the cohomology of the graph complex with coefficients in a theory. The Kontsevich-type invariant is then an evaluation, on a concrete cyclic algebra, of this class. We also explain some results of M. Penkava and A. Schwarz on the construction of an invariant from a cyclic deformation of a cyclic algebra. Our constructions are illustrated by a `toy model' of tree complexes.Comment: LaTeX 2.09 + article12pt,leqno style, 10 page

Homotopy groups of Hom complexes of graphs

The notion of $\times$-homotopy from \cite{DocHom} is investigated in the context of the category of pointed graphs. The main result is a long exact sequence that relates the higher homotopy groups of the space \Hom_*(G,H) with the homotopy groups of \Hom_*(G,H^I). Here \Hom_*(G,H) is a space which parametrizes pointed graph maps from $G$ to $H$ (a pointed version of the usual \Hom complex), and $H^I$ is the graph of based paths in $H$. As a corollary it is shown that \pi_i \big(\Hom_*(G,H) \big) \cong [G,\Omega^i H]_{\times}, where $\Omega H$ is the graph of based closed paths in $H$ and $[G,K]_{\times}$ is the set of $\times$-homotopy classes of pointed graph maps from $G$ to $K$. This is similar in spirit to the results of \cite{BBLL}, where the authors seek a space whose homotopy groups encode a similarly defined homotopy theory for graphs. The categorical connections to those constructions are discussed.Comment: 20 pages, 6 figures, final version, to be published in J. Combin. Theory Ser.