SU(2)-abelian graph manifolds with a single JSJ torus

TL;DR

This paper classifies SU(2)-abelian graph manifolds with a single JSJ torus, characterized by Seifert coefficients, and shows they are Heegaard Floer homology L-spaces, advancing understanding of their geometric and topological properties.

Contribution

It provides a classification of SU(2)-abelian graph manifolds with one JSJ torus using Seifert coefficients and proves they are L-spaces in Heegaard Floer homology.

Findings

Classification of SU(2)-abelian graph manifolds with a single JSJ torus.

Identification of these manifolds as L-spaces in Heegaard Floer homology.

Abstract

A 3-manifold is called \emph{SU(2)}-abelian if every SU(2)-representation of its fundamental group has abelian image. We classify, in terms of the Seifert coefficients, SU(2)-abelian 3-manifolds among the family of graph manifolds obtained by gluing two Seifert spaces both fibred over a disk and with two singular fibers. Finally, we prove that these SU(2)-abelian manifolds are Heegaard Floer homology L-spaces.