Generalized Dijkgraaf-Witten theories and stable diffeomorphism types of 4-manifolds, II

Topological field theories (TFTs) provide diffeomorphism invariants of manifolds. A central question is what sorts of manifold invariants arise from TFTs? When is it possible to distinguish distinct manifolds by TFTs? In this second part of our series, I will discuss joint work with Christopher Schommer-Pries outlining an answer to this question for even-dimensional semisimple TFTs. Manifolds satisfying a certain finiteness condition are indistinguishable by such field theories precisely if they are stably diffeomorphic. 

 

After recalling the notions of semisimple TFT and stable diffeomorphism, I will construct a family of semisimple TFTs which generalize a classical construction of Dijkgraaf and Witten. These field theories are built from spaces over BO(n) and can be thought of as finite sigma models, or finite gauge theories. I will explain why these field theories precisely see the stable diffeomorphism type of sufficiently finite even-dimensional manifolds.