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

We define Pin(2)-equivariant Seiberg-Witten Floer homology for rational homology 3-spheres equipped with a spin structure. The analogue of Froyshov'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 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.Comment: 29 pages; final version, to appear in Journal of the AM

Lectures on the triangulation conjecture

We outline the proof that non-triangulable manifolds exist in any dimension greater than four. The arguments involve homology cobordism invariants coming from the Pin(2) symmetry of the Seiberg-Witten equations. We also explore a related construction, of an involutive version of Heegaard Floer homology.Comment: 33 pages. Notes prepared with the help of Eylem Zeliha Yildiz; to appear in Proceedings of the 22nd Gokova Geometry/Topology Conference. The arxiv version has a corrected statement on p.