The geometric cobordism hypothesis

Abstract

We prove a generalization of the cobordism hypothesis of Baez--Dolan and Hopkins--Lurie for bordisms with arbitrary geometric structures, such as Riemannian metrics, complex and symplectic structures, principal bundles with connections, or geometric string structures. Our methods rely on the locality property for fully extended functorial field theories established in arXiv:2011.01208, reducing the problem to the special case of geometrically framed bordism categories. As an application, we upgrade the classification of invertible fully extended topological field theories by B\"okstedt--Madsen and Schommer-Pries to nontopological field theories, generalizing the work of Galatius--Madsen--Tillmann--Weiss to arbitrary geometric structures.