Pin(2)-Equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture

We define Pin ⁡ ( 2 ) Pin(2) -equivariant Seiberg-Witten Floer homology for rational homology 3 3 -spheres equipped with a spin structure. The analogue of Frøyshov’s correction term in this setting is an integer-valued invariant of homology cobordism whose mod 2 2 reduction is the Rokhlin invariant. As an application, we show that there are no homology 3 3 -spheres Y Y of the Rokhlin invariant one such that Y # Y Y #Y bounds an acyclic smooth 4 4 -manifold. By previous work of Galewski-Stern and Matumoto, this implies the existence of non-triangulable high-dimensional manifolds.