An A(infinity)Operad in Spineless Cacti

Abstract

The dg operad of cellular chains on the operad of spineless cacti of Kaufmann (Topology 46(1):39-88, 2007) is isomorphic to the Gerstenhaber-Voronov dg operad codifying the cup product and brace operations on the Hochschild cochains of an associative algebra, and to the suboperad of the surjection operad of Berger and Fresse (Math Proc Camb Philos Soc 137(1):135-174, 2004), McClure and Smith (Recent progress in homotopy theory (Baltimore, MD, 2000). Contemp Math., Amer. Math. Soc., Providence 293:153-193, 2002) and McClure and Smith (J Am Math Soc 16(3):681-704, 2003). Its homology is the Gerstenhaber dg operad . We construct a map of dg operads such that is commutative and is the canonical map . This formalises the idea that, since the cup product is commutative in homology, its symmetrisation is a homotopy associative operation. Our explicit structure does not vanish on non-trivial shuffles in higher degrees, so does not give a map . If such a map could be written down explicitly, it would immediately lead to a structure on and on Hochschild cochains, that is, to an explicit and direct proof of the Deligne conjecture.