Constructing Simplicial Branched Covers

TL;DR

This paper explores the construction of simplicial branched covers, demonstrating that every closed oriented PL d-manifold for d<=4 can be realized as a partial unfolding of a polytopal d-sphere, expanding the toolkit for topological constructions.

Contribution

It introduces a broad class of branched covers constructed through partial unfolding, showing their applicability to all closed oriented PL d-manifolds for dimensions up to four.

Findings

Every closed oriented PL 4-manifold is a partial unfolding of some polytopal 4-sphere.

The partial unfolding method can generate a wide class of simplicial branched covers.

The approach extends known bounds and constructions in low-dimensional topology.

Abstract

Branched covers are applied frequently in topology - most prominently in the construction of closed oriented PL d-manifolds. In particular, strong bounds for the number of sheets and the topology of the branching set are known for dimension d<=4. On the other hand, Izmestiev and Joswig described how to obtain a simplicial covering space (the partial unfolding) of a given simplicial complex, thus obtaining a simplicial branched cover [Adv. Geom. 3(2):191-255, 2003]. We present a large class of branched covers which can be constructed via the partial unfolding. In particular, for d<=4 every closed oriented PL d-manifold is the partial unfolding of some polytopal d-sphere.