On Spineless Cacti, Deligne’s Conjecture and Connes–kreimer’s Hopf Algebra

We give a new direct proof of Deligne’s conjecture on the Hochschild cohomology. For this we use the cellular chain operad of normalized spineless cacti as a model for the chains of the little discs operad. Previously, we had shown that the operad of spineless cacti is homotopy equivalent to the little discs operad. Moreover, we also showed that the quasi–operad of normalized spineless cacti is homotopy equivalent to the spineless cacti operad. Now, we give a cell decomposition for the normalized spineless cacti, whose cellular chains form an operad. The cells are indexed by bipartite black and white trees which can directly be interpreted as operations on the Hochschild cochains of an associative algebra. Furthermore, the symmetric combinations of top–dimensional cells, are identified as the pre–Lie operad whose Hopf algebra as an operad is the renormalization Hopf algebra of Connes and Kreimer.