The geometric cobordism hypothesis

Understand

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.