Kevin Walker, Microsoft Research

Four reasons why 1- and 2-manifolds are more interesting than you thought

Abstract: What surfaces admit a relative Morse function without any critical points? The unexciting answer: Only mapping cylinders. But if we glue several mapping cylinders together along codimension 0 submanifolds of their boundaries, we eventually arrive at an interesting colored operad which enlarges the little disks operad. This new operad leads to generalizations of some familiar constructions, including the Deligne conjecture for Hochschild cohomology and the Drinfeld center and double (together with the associated braidings and twists). None of these generalizations requires increasing the category number or placing any additional requirements on the input algebras or tensor categories.