Defining extended TQFTs via handle attachments

TL;DR

This paper provides a finite presentation of the bicategory of cobordisms with corners using handle attachments, enabling the construction of extended TQFTs from categorified TQFTs with handle relations.

Contribution

It introduces a finite set of generators and relations for the cobordism bicategory, extending the framework of TQFTs to include handle attachments and their invariances.

Findings

Finite presentation of the cobordism bicategory.

Construction of extended TQFTs from categorified TQFTs.

Handle attachment relations ensure invariance and extendability.

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.