Crossed interval groups and operations on the Hochschild cohomology

Abstract

We prove that the operad $\mathcal{B}$ 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 $\mathcal{B}$ is a certain crossed interval extension of an operad $\mathcal{T}$ whose homotopy type is known. This completes the investigation of the algebraic structure on the Hochschild cochain complex that has lasted for several decades.