Hochschild cohomology and string topology of global quotient orbifolds

TL;DR

This paper establishes a direct algebraic connection between the string topology of orbifolds formed by a finite group acting on a manifold and the Hochschild cohomology of a related dg-ring, deepening understanding of orbifold invariants.

Contribution

It provides an explicit ring isomorphism linking orbifold string topology with Hochschild cohomology of a constructed dg-ring, bridging geometric and algebraic perspectives.

Findings

Ring isomorphism between orbifold string topology and Hochschild cohomology.

Explicit description of the dg-ring involving the smash product.

Enhanced understanding of orbifold invariants through algebraic methods.

Abstract

Let M be a connected, simply connected, closed and oriented manifold, and G a finite group acting on M by orientation preserving diffeomorphisms. In this paper we show an explicit ring isomorphism between the orbifold string topology of the orbifold [M/G] and the Hochschild cohomology of the dg-ring obtained by performing the smash product between the group G and the singular cochain complex of M.