When determining the fundamental group of the circle in Sect.6.4, we considered the exponential map, which laid out the real numbers like a helix over the circle and thus ‘covered’ it. The maps we will consider in this chapter are generalisations of this situation. The lifting behaviour of paths in coverings can be used to calculate fundamental groups. The connection between the fundamental group and coverings is even closer and leads to the classification of coverings in terms of the fundamental group. The whole theory is analogous to the Galois theory of field extensions.

Authors and Affiliations

1. Fakultät für Mathematik, Ruhr-Universität Bochum, Bochum, Germany

   Gerd Laures

2. School of Mathematical and Physical Sciences, University of Sheffield, Sheffield, UK

   Markus Szymik

Reprints and permissions

© 2025 The Author(s), under exclusive license to Springer-Verlag GmbH, DE, part of Springer Nature

Policies and ethics