# An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves

IIT Madras Course , Prof. T.E. Venkata Balaji

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### Lecture 15: Completion of the Construction of the Universal Coveringl

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# Lecture Details

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 httpwww.nptel.iitm.ac.insyllabus111106044Goals of Lecture 15 Part A 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 this lecture, we show that the universal covering space we constructed is indeed simply connected and has a universal property We show that the universal covering space we have constructed is also a covering space for any other covering space. We further show that any covering space which is simply connected is homeomorphic to the universal covering space we have constructed. It follows that any two simply connected covering spaces thereby are not only just homeomorphic, but homeomorphic by a map that respects the covering projections, i.e., are isomorphic as covering spaces; in fact, even the isomorphism becomes unique if a point of the source and one of the target are fixed. These results show the universality of a simply connected covering space, which is why such a space is called "the" universal covering spaceKeywords for Lecture 15 Part A 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

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