Combinatorial Characterizations and Branched Manifolds
Daryl Cooper, Leslie Mavrakis, Priyam Patel
TL;DR
The paper develops a combinatorial framework for families of compact $n$-manifolds defined by local structure, introducing LCD (locally combinatorially defined) and proving its equivalence to the geometric notion of immersion into a compact PL branched $n$-manifold (BM). It provides a constructive path to universal branched manifolds: starting from finite local models and labeling schemes (colors and geographies), it shows how to encode local data into finite model sets and build a universal branched object that represents the entire LCD family. Key results include the Equivalence Theorem LCD ⇔ BM and the existence of universal branched manifolds for geometric families, along with explicit applications to torus bundles over the circle and implications for closed orientable 3-manifolds in Thurston geometries. The framework offers a finite-model, combinatorial approach to broad classes of geometric-topology questions, enabling universal representations across different geometric structures and paving the way for systematic explorations of 3-manifold geometries via branched immersion theory.
Abstract
A family of compact n-manifolds is locally combinatorially defined (LCD) if it can be specified by a finite number of local triangulations. We show that LCD is equivalent to the existence of a compact branched n-manifold W, such that the family is precisely those manifolds that immerse into W. In subsequent papers, the equivalence will be used to show that, for each of the eight Thurston geometries, the family of closed 3-manifolds admitting that geometry is LCD.