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

Manolescu, Ciprian

doi:10.1090/jams829

Pin(2)-Equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture
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J. Amer. Math. Soc. 29 (2016), 147-176 Request permission

Abstract:

We define $\operatorname {Pin}(2)$-equivariant Seiberg-Witten Floer homology for rational homology $3$-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 $2$ reduction is the Rokhlin invariant. As an application, we show that there are no homology $3$-spheres $Y$ of the Rokhlin invariant one such that $Y \#Y$ bounds an acyclic smooth $4$-manifold. By previous work of Galewski-Stern and Matumoto, this implies the existence of non-triangulable high-dimensional manifolds.

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