Introduction to orbifolds

TL;DR

This paper introduces orbifolds using classical charts and explores their fundamental topological, geometric, and algebraic properties, including generalizations of classical theorems across various mathematical fields.

Contribution

It provides a comprehensive introduction to orbifolds, detailing their structures and extending fundamental concepts from topology, geometry, and algebra to this setting.

Findings

Orbifolds generalize classical manifolds with singularities.

Fundamental groups and coverings are adapted for orbifolds.

Classical theorems are extended to orbifold contexts.

Abstract

We introduce orbifolds from the classical point of view, using charts, and present orbifold versions of elementary objects from Algebraic Topology, such as the fundamental group, coverings and Euler characteristic; Differential Topology/Geometry, including orbibundles, differential forms, integration and (equivariant) De Rham cohomology; and Riemannian Geometry, surveying generalizations of classical theorems to this setting.