Completion of the Construction of the Universal Covering: The Fundamental Group

An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/

Goals of Lecture 15 Part B:

* In the previous couple of lectures, the universal covering space was constructed for a given space as a Hausdorff topological space along with a natural map into the given space. That natural map was shown to be a covering map. In the first part (part A) of this lecture, we showed that the universal covering space we constructed is indeed simply connected and has a universal property. In this lecture (part B), we show that we can naturally identify the fundamental group of the base space with a subgroup of self-isomorphisms of the universal covering space called the Deck Transformation Group

* For any covering space, we may define the so-called Deck Transformation Group. This is the subgroup of self-homeomorphisms of the covering space that respect the covering projection map. If the covering space is the universal covering space, then the fundamental group of the base space (the space whose coverings we are concerned with) gets naturally identified with the deck transformation group.
Thus the fundamental group of the base acts on the universal covering via the so-called deck transformations. These act along the fibers of the covering projection map. This action is called the Monodromy Action


Keywords for Lecture 15 Part B:

Path, Fixed-end-point (FEP) homotopy equivalence class, fundamental group, pathwise or arcwise connected, Hausdorff, locally simply connected, universal covering, basic open set, base for a topology, sub-base for a topology, admissible neighborhood, isomorphism of covering spaces, universal property, deck transformation, deck transformation group, monodromy action