A Comparison of Products in Hochschild Cohomology

TL;DR

This paper explores the relationship between Steenrod's cup-i products and Hochschild cohomology, establishing equivalences and implications for algebraic structures like Gerstenhaber's products and Batalin-Vilkovisky structures.

Contribution

It demonstrates how Steenrod's cup-i products can be transported to Hochschild cochains, linking topological and algebraic cohomology operations in a novel way.

Findings

Gerstenhaber's cup product matches the simplicial cup product for cocycles supported on BG.

Gerstenhaber's pre-Lie product agrees with Steenrod's cup-one product.

The Gerstenhaber bracket vanishes for cocycles supported on BG.

Abstract

We transport Steenrod's cup-i products from the singular cochains on the free loop space Maps(S^1, BG) to Hochschild's original cochain complex Hom (k[G]^*, k[G]) defining Hochschild cohomology. Here G is a discrete group, k an arbitrary coefficient ring, and BG the classifying space of G. For cochains supported on BG, we prove that Gerstenhaber's cup product agrees with the simplicial cup product and Gerstenhaber's pre-Lie product agrees with Steenrod's cup-one product. As a consequence, for cocycles f and g supported on BG, the Gerstenhaber bracket [f, g] = 0 in Hochschild cohomology. This is interpreted in terms of the Batalin-Vilkovisky structure on the Hochschild cohomology of k[G].