Elementary Orbifold Differential Topology

TL;DR

This paper introduces a simple approach to orbifold differential topology, establishing foundational results like Sard's theorem and exploring implications for orbifold maps and isotropy groups.

Contribution

It develops the concept of regular values for smooth orbifold maps and proves key theorems, including Sard's theorem and a Borsuk no retraction theorem, in the orbifold setting.

Findings

Sard's theorem holds for smooth orbifold maps.

Inverse images of regular values are smooth suborbifolds.

Obstructions to real-valued orbifold maps are identified.

Abstract

Taking an elementary and straightforward approach, we develop the concept of a regular value for a smooth map f: O -> P between smooth orbifolds O and P. We show that Sard's theorem holds and that the inverse image of a regular value is a smooth full suborbifold of O. We also study some constraints that the existence of a smooth orbifold map imposes on local isotropy groups. As an application, we prove a Borsuk no retraction theorem for compact orbifolds with boundary and some obstructions to the existence of real-valued orbifold maps from local model orbifold charts.