Characterization of principal bundles: the commutative case

TL;DR

This paper reviews the characterization of principal bundles focusing on group actions and maps across three stages involving topological, discrete, and Lie groups acting on various spaces.

Contribution

It provides a comprehensive overview of principal bundle characterization across different types of groups and spaces, highlighting the properties of group actions and related maps.

Findings

Unified framework for principal bundle characterization

Analysis of group actions in topological and smooth contexts

Clarification of canonical and translation maps in principal bundles

Abstract

A review of the characterization of principal bundles, through the different properties of the action of a group and its related canonical and translation maps, is presented. The work is divided in three stages: a topological group acting on a topological space, a discrete group acting on a smooth manifold, and a Lie group acting on a smooth manifold.