From loom spaces to veering triangulations

TL;DR

This paper introduces loom spaces, generalizing structures related to pseudo-Anosov flows and veering triangulations, and proves a canonical association with locally veering triangulations homeomorphic to -space.

Contribution

It establishes a new framework of loom spaces and links them to veering triangulations, expanding understanding of 3-manifold structures.

Findings

Loom spaces generalize leaf and link spaces.

Every loom space has a canonical associated veering triangulation.

The realization of this triangulation is homeomorphic to .

Abstract

We introduce loom spaces, a generalisation of both the leaf spaces associated to pseudo-Anosov flows and the link spaces associated to veering triangulations. Following work of Gu\'eritaud, we prove that there is a locally veering triangulation canonically associated to every loom space, and that the realisation of this triangulation is homeomorphic to $\mathbb{R}^3$.