This talk will discuss generalized Dijkgraaf-Witten topological field theories and their ability to detect stable diffeomorphism classes of manifolds. Like a gemstone, there are many facets to explore. Dijkgraaf-Witten theories belong to a class of topological field theories obtained by “finite path integration”. The categorical underpinnings of this construction, which could be called “rational ambidexterity”, allow one to take highly structured TQFTs and integrate out the structure. The classification of invertible TQFTs in terms of stable homotopy gives a rich source of inputs for this construction. Then there is Kreck’s classification of manifolds up to stable diffeomorphism. I will try to touch on these topics, explain how they come together, and what they tell us about TQFTs.