An Introduction to Riemann Surfaces pp 477-530 | Cite as

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

## Abstract

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 *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.

## Keywords

Fundamental Group Cohomology Group Homology Group Deck Transformation Countable Surface## References

- [Fa]M. Farber,
*Topology of Closed One-Forms*, Mathematical Surveys and Monographs, 108, American Mathematical Society, Providence, 2004. MATHGoogle Scholar - [For]O. Forster,
*Lectures on Riemann Surfaces*, Graduate Texts in Mathematics, 81, Springer, Berlin, 1981. MATHCrossRefGoogle Scholar - [Hat]A. Hatcher,
*Algebraic Topology*, Cambridge University Press, Cambridge, 2001. Google Scholar - [Wa]F. Warner,
*Foundations of Differentiable Manifolds and Lie Groups*, Graduate Texts in Mathematics, 94, Springer, New York, 1983. MATHCrossRefGoogle Scholar - [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