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

TL;DR

This paper introduces a new Pin(2)-equivariant Seiberg-Witten Floer homology for rational homology 3-spheres with spin structures, providing tools to address the Triangulation Conjecture through novel invariants.

Contribution

It defines a Pin(2)-equivariant Floer homology and an associated integer-valued invariant, linking it to the Rokhlin invariant and high-dimensional manifold triangulation problems.

Findings

No homology 3-spheres with Rokhlin invariant one bound an acyclic 4-manifold.

The invariant generalizes Froyshov's correction term to the Pin(2) setting.

Implications for the existence of non-triangulable manifolds in high dimensions.

Abstract

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.