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We define Pin(2)-equivariant Seiberg-Witten Floer homology for rationalhomology 3-spheres equipped with a spin structure. The analogue of Froyshov'scorrection term in this setting is an integer-valued invariant of homologycobordism whose mod 2 reduction is the Rokhlin invariant. As an application, weshow that there are no homology 3-spheres Y of Rokhlin invariant one such thatY # Y bounds an acyclic smooth 4-manifold. By previous work of Galewski-Sternand Matumoto, this implies the existence of non-triangulable high-dimensionalmanifolds.

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