An A-infinity operad in spineless cacti

TL;DR

This paper constructs an A-infinity operad map in the context of spineless cacti, linking it to Hochschild cochains and the Deligne conjecture, and providing explicit algebraic structures that formalize homotopy associative operations.

Contribution

It explicitly constructs an A-infinity operad map in the spineless cacti operad, connecting it to Hochschild cochains and the Gerstenhaber operad, advancing understanding of homotopy algebra structures.

Findings

Constructed an explicit A-infinity operad map psi in the spineless cacti operad.

Showed that the homology of the operad map corresponds to the Gerstenhaber operad.

Highlighted the potential for a G-infinity structure on Hochschild cochains and implications for the Deligne conjecture.

Abstract

The d.g. operad C of cellular chains on the operad of spineless cacti is isomorphic to the Gerstenhaber-Voronov operad codifying the cup product and brace operations on the Hochschild cochains of an associative algebra, and to the suboperad F_2X of the surjection operad. Its homology is the Gerstenhaber operad G. We construct an operad map psi from A-infinity to C such that psi(m_2) is commutative and the homology of psi is the canonical map A \to Com \to G. This formalises the idea that, since the cup product is commutative in homology, its symmetrisation is a homotopy associative operation. Our explicit A-infinty structure does not vanish on non-trivial shuffles in higher degrees, so does not give a map from Com-infinity to C. If such a map could be written down explicitly, it would immediately lead to a G-infinity structure on C and on Hochschild cochains, that is, to a direct proof of the Deligne conjecture.