An ${\mathcal{A}_\infty}$ Operad in Spineless Cacti

Abstract

The dg operad ${\mathcal{C}}$ 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 ${F_2\mathcal{X}}$ 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 ${\mathcal{G}}$. We construct a map of dg operads ${\psi \colon \mathcal{A}_\infty \longrightarrow \mathcal{C}}$ such that ${\psi(m_2)}$ is commutative and ${H_*(\psi)}$ is the canonical map ${\mathcal{A} \to \mathcal{C}\!om \to \mathcal{G}}$. This formalises the idea that, since the cup product is commutative in homology, its symmetrisation is a homotopy associative operation. Our explicit ${\mathcal{A}_\infty}$ structure does not vanish on non-trivial shuffles in higher degrees, so does not give a map ${\mathcal{C}om_\infty \to \mathcal{C}}$. If such a map could be written down explicitly, it would immediately lead to a ${\mathcal{G}_\infty}$ structure on ${\mathcal{C}}$ and on Hochschild cochains, that is, to an explicit and direct proof of the Deligne conjecture.