Natural symmetries of secondary Hochschild homology

TL;DR

This paper explores the symmetries of twice-iterated Hochschild homology through the lens of framed diffeomorphisms of the torus, revealing connections to braid groups and proposing a new secondary cyclotomic structure.

Contribution

It identifies the framed diffeomorphism group of the torus as a semi-direct product involving braid groups and introduces an unstable secondary cyclotomic structure for iterated Hochschild homology.

Findings

Framed diffeomorphisms form a semi-direct product with braid groups.

The action of these symmetries extends to framed local-diffeomorphisms in Cartesian monoidal structures.

Proposes a new unstable secondary cyclotomic structure for iterated Hochschild homology.

Abstract

We identify the group of framed diffeomorphisms of the torus as a semi-direct product of the torus with the braid group on 3 strands; we also identify the topological monoid of framed local-diffeomorphisms of the torus in similar terms. It follows that the framed mapping class group is this braid group. We show that the group of framed diffeomorphisms of the torus acts on twice-iterated Hochschild homology, and explain how this recovers a host of familiar symmetries. In the case of Cartesian monoidal structures, we show that this action extends to the monoid of framed local-diffeomorphisms of the torus. Based on this, we propose a definition of an unstable secondary cyclotomic structure, and show that iterated Hochschild homology possesses such in the Cartesian monoidal setting.