Hochschild cohomology of intersection complexes and Batalin-Vilkovisky algebras

Abstract

Let $X$ be a compact, oriented, second countable pseudomanifold. We show that $HH^\ast_\bullet(\widetilde N^\ast_\bullet(X;\mathbb{Q}))$, the Hochschild cohomology of the blown-up intersection cochain complex of $X$, is well defined and endowed with a Batalin-Vilkovisky algebra structure. Furthermore, we prove that it is a topological invariant. More generally, we define the Hochschild cohomology of a perverse differential graded algebra $A_\bullet$ and present a natural Gerstenhaber algebra structure on it. This structure can be extended into a Batalin-Vilkovisky algebra when $A_\bullet$ is a derived Poincar\'e duality algebra.