Graph manifolds, left-orderability and amalgamation

TL;DR

This paper proves that all irreducible toroidal integer homology sphere graph manifolds have left-orderable fundamental groups, combining results from 3-manifold topology, Heegaard Floer homology, and amalgamation theory.

Contribution

It establishes a new class of 3-manifolds with left-orderable fundamental groups using a specialized amalgamation approach.

Findings

All irreducible toroidal integer homology sphere graph manifolds are left-orderable.

The proof relies on a specialization of Bludov and Glass's result for amalgamated products.

The result integrates techniques from 3-manifold topology and Heegaard Floer homology.

Abstract

We show that every irreducible toroidal integer homology sphere graph manifold has a left-orderable fundamental group. This is established by way of a specialization of a result due to Bludov and Glass for the almagamated products that arise, and in this setting work of Boyer, Rolfsen and Wiest may be applied. Our result then depends on input from 3-manifold topology and Heegaard Floer homology.