Toroidal homology spheres and SU(2)-representations

TL;DR

This paper proves that certain three-dimensional homology spheres with embedded tori have fundamental groups admitting irreducible SU(2)-representations, using advanced tools from instanton Floer homology and surgery theory.

Contribution

It establishes a new link between the topology of homology spheres with tori and the existence of irreducible SU(2)-representations of their fundamental groups, employing novel Floer homology techniques.

Findings

Embedded incompressible tori imply irreducible SU(2)-representations

Uses instanton Floer homology and surgery exact triangle methods

Connects topology of homology spheres to group representations

Abstract

We prove that if an integer homology three-sphere contains an embedded incompressible torus, then its fundamental group admits irreducible SU(2)-representations. Our methods use instanton Floer homology, and in particular the surgery exact triangle, holonomy perturbations, and a non-vanishing result due to Kronheimer-Mrowka, as well as results about surgeries on cables due to Gordon.