Semisimple Field Theories Detect Stable Diffeomorphism
David Reutter, Christopher Schommer-Pries
TL;DR
This work develops semisimple topological field theories (TFTs) in arbitrary even dimensions and proves they detect only stable diffeomorphism classes, establishing a robust upper bound on manifold invariants for this class. It introduces finite path integral theories as a higher-categorical generalization of Dijkgraaf–Witten theories, showing these are semisimple and, under finiteness assumptions, optimally distinguish manifolds up to stable diffeomorphism via type-(q−1) theories. The authors develop a comprehensive framework based on spans, local systems, and Pontryagin pairings to characterize what finite path integral theories can see, including a nondegeneracy theory and a precise reduction to tangential n-types and bordism orbits. They connect these invariants to Kreck’s stable diffeomorphism classification, demonstrate dimensional reduction preserves finiteness, and illustrate applications to unoriented 4-manifolds and exotic spheres, while providing a bicategorical extension for invertible theories. Altogether, the paper unifies tangential bordism, finite groupoid-like path integrals, and stable diffeomorphism theory to show semisimple TFTs capture stable diffeomorphism data and to delineate the reach of finite path integral invariants.
Abstract
Extending the work of the first author, we introduce a notion of semisimple topological field theory in arbitrary even dimension and show that such field theories necessarily lead to stable diffeomorphism invariants. The main result of this paper is a proof that this 'upper bound' is optimal: To this end, we introduce and study a class of `finite path integral' topological field theories which are semisimple and which generalize well known theories constructed by Dijkgraaf-Witten, Freed and Quinn. We show that manifolds satisfying a certain finiteness condition -- including 4-manifolds with finite fundamental group -- are indistinguishable to these field theories if and only if they are stably diffeomorphic. Subject to these finiteness conditions, such finite path integral theories therefore provide the strongest semisimple TFT invariants possible. These results hold for a large class of ambient tangential structures. We discuss a number of applications, including the constructions of unoriented 4-dimensional semisimple field theories which can distinguish unoriented smooth structure and oriented higher-dimensional semisimple field theories which can distinguish certain exotic spheres. Along the way, we show that dimensional reductions of finite path integral theories are again finite path integral theories, we utilize ambidexterity in the rational setting, and we develop techniques related to the $\infty$-categorical Möbius inversion principle of Gálvez-Carrillo--Kock--Tonks.