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

Part of the Cornerstones book series (COR)


We 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 pX 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.


Fundamental Group Cohomology Group Homology Group Deck Transformation Countable Surface 
These 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.


  1. [Fa]
    M. Farber, Topology of Closed One-Forms, Mathematical Surveys and Monographs, 108, American Mathematical Society, Providence, 2004. MATHGoogle Scholar
  2. [For]
    O. Forster, Lectures on Riemann Surfaces, Graduate Texts in Mathematics, 81, Springer, Berlin, 1981. MATHCrossRefGoogle Scholar
  3. [Hat]
    A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2001. Google Scholar
  4. [Wa]
    F. Warner, Foundations of Differentiable Manifolds and Lie Groups, Graduate Texts in Mathematics, 94, Springer, New York, 1983. MATHCrossRefGoogle Scholar
  5. [Wey]
    H. Weyl, The Concept of a Riemann Surface, translated from the third German ed. by Gerald R. MacLane, International Series in Mathematics, Addison-Wesley, Reading, 1964. Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsLehigh UniversityBethlehemUSA
  2. 2.Department of MathematicsSUNY at BuffaloBuffaloUSA

Personalised recommendations