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

By Ciprian Manolescu. 2013

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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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/08c984c967297486a321603b771572839f9a408f

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Citations in the corpus

Connected sums and involutive knot Floer homology (2017, background): “Monopole Floer homology is an S(1)-equivariant theory, however the action of S(1) can be enlarged to an action of Pin(2), and there are now constructions which take into account the full Pin(2) action [20, 22].”
Involutive Heegaard Floer homology (2015, methodology): “The spectrum SWF (Y ) comes equipped with a Pin(2) action extending the S1 action, and its Pin(2)-equivariant homology was studied in [24].”
A survey of the foundations of four-manifold theory in the topological category (2019, methodology): “This question was reduced to a problem about homology 3-spheres [Mat78, GS80], which was then solved by Manolescu [Man16].”
A connected sum formula for involutive Heegaard Floer homology (2016, background): “We recall that the group Θ(3)Z plays a fundamental role in the study of triangulations on high dimensional manifolds; see [5, 17, 15].”