In this chapter, we define the fundamental group of a (pointed) space as the set of path components of the corresponding loop space and study its properties. Initially, we can imagine that maps from the circle\(S^1\) into a topological spaceX (that map 1 to a fixed pointx) always have a ‘generalised mapping degree’. However, it does not take values in\(\mathbb {Z}\) but in the fundamental group\(\pi _1(X,x)\). We will develop techniques for calculating these fundamental groups by characterising their behaviour with respect to coverings. Subsequently, we illustrate the results in the case of surfaces.

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

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