Background Material on Fundamental Groups, Covering Spaces, and (Co)homology

Abstract

AbstractWe recall that a path (or parametrized path or curve or parametrized curve) in a topological space X from a point x to a point y is a continuous mapping γ:[a,b]→X with γ(a)=x and γ(b)=y (see Sect. 9.1). We take the domain of a path to be [0,1], unless otherwise indicated. A loop (or closed curve) with base point p∈X is a path in X from p to p. In this chapter, we consider the equivalence relation given by path homotopies. This leads to the fundamental group, which is the group given by the path homotopy equivalence classes of loops at a point, and to covering spaces, both of which are important objects in complex analysis and Riemann surface theory. We also consider homology groups, which are essentially Abelian versions of the fundamental group, and cohomology groups, which are groups that are dual to the homology groups.KeywordsFundamental GroupCohomology GroupHomology GroupDeck TransformationCountable SurfaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.