Kevin Walker, Microsoft Research
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Four reasons why 1- and 2-manifolds are more interesting than you thought
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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.