The Diffeomorphism Group of the Solid Closed Torus and Hochschild Homology

TL;DR

This paper demonstrates that in a specific categorical setting, the cyclic action on Hochschild homology extends to an action of the diffeomorphism group of the solid closed torus, revealing a deep connection between category theory and 3-manifold topology.

Contribution

It establishes a new extension of the cyclic action on Hochschild complexes to the diffeomorphism group of the solid torus within a particular categorical framework.

Findings

Cyclic action on Hochschild complex extends to diffeomorphism group

Connects categorical Hochschild homology with 3-manifold topology

Provides new insights into the structure of ribbon Grothendieck-Verdier categories

Abstract

We prove that for a self-injective ribbon Grothendieck-Verdier category $\mathcal{C}$ in the sense of Boyarchenko-Drinfeld the cyclic action on the Hochschild complex of $\mathcal{C}$ extends to an action of the diffeomorphism group of the solid closed torus $\mathbb{S}^1 \times \mathbb{D}^2$.