Crossed interval groups and operations on the Hochschild cohomology

TL;DR

This paper establishes a deep connection between the operad of natural operations on Hochschild cohomology and the little disks operad by introducing crossed interval groups, completing a long-standing investigation.

Contribution

It introduces crossed interval groups and shows that the operad of natural operations on Hochschild cohomology has the homotopy type of the little disks operad.

Findings

Operad B has the homotopy type of chains on the little disks operad.

Crossed interval groups are used to relate operad B to a known operad T.

The work completes a decades-long study of algebraic structures on Hochschild cochains.

Abstract

We prove that the operad B of natural operations on the Hochschild cohomology has the homotopy type of the operad of singular chains on the little disks operad. To achieve this goal, we introduce crossed interval groups and show that B is a certain crossed interval extension of an operad T whose homotopy type is known. This completes the investigation of the algebraic structure on the Hochschild cochain complex that has lasted for several decades.