Genus one open books with non-left-orderable fundamental group

TL;DR

This paper proves that for certain genus one open book decompositions of three-manifolds, being an L-space implies the fundamental group is not left-orderable, addressing a question in 3-manifold topology.

Contribution

It establishes a link between L-space property and non-left-orderability of the fundamental group for genus one open books, answering a previously open question.

Findings

L-space manifolds with genus one open books have non-left-orderable fundamental groups.

The result confirms a conjecture relating L-spaces and orderability in this class.

Provides new insights into the structure of 3-manifolds with genus one open books.

Abstract

Let $Y$ be a closed, connected, orientable three-manifold admitting a genus one open book decomposition with one boundary component. We prove that if $Y$ is an L-space, then the fundamental group of $Y$ is not left-orderable. This answers a question posed by John Baldwin.