Defining extended TQFTs via handle attachments

Abstract

We give a finite presentation of the cobordism symmetric monoidal bicategory of (smooth, oriented) closed manifolds, cobordisms and cobordisms with corners as an extension of the bicategory of closed manifolds, cobordisms and diffeomorphisms. The generators are the standard handle attachments, and the relations are handle cancellations and invariance under reversing the orientation of the attaching spheres. In other words, given a categorified TQFT and 2-morphisms associated to the standard handles satisfying our relations, we construct a once extended TQFT.

v2: Many figures added. To appear in Algebraic and Geometric Topology. Comments welcome!