The Hochschild cohomology ring of the singular cochain algebra of a space

TL;DR

This paper computes the algebraic structure of Hochschild cohomology for singular cochain algebras on spaces with polynomial or exterior algebra cohomology, including spectral sequence and Batalin-Vilkovisky structures.

Contribution

It provides explicit descriptions of Hochschild cohomology rings and BV algebra structures for specific classes of topological spaces, advancing understanding in algebraic topology.

Findings

Hochschild cohomology ring structure determined for polynomial cohomology spaces.

Spectral sequence calculations of Hochschild cohomology presented.

Batalin-Vilkovisky algebra structure identified in characteristic two cases.

Abstract

We determine the algebra structure of the Hochschild cohomology of the singular cochain algebra with coefficients in a field on a space whose cohomology is a polynomial algebra. A spectral sequence calculation of the Hochschild cohomology is also described. In particular, when the underlying field is of characteristic two, we determine the associated bigraded Batalin-Vilkovisky algebra structure on the Hochschild cohomology of the singular cochain on a space whose cohomology is an exterior algebra.