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 10: The Importance of the Path-lifting Property

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        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 10 To explore the reasons for the fundamental group occurring both as the inverse image of any point under the universal covering map as well as a subgroup of automorphisms of the universal covering space To understand the notions of lifting property, unique-lifting property and uniqueness-of-lifting property To understand the Covering Homotopy Theorem To note that surjective local homeomorphisms have the uniqueness-of-lifting property To note that a surjective local homeomorphism is a covering iff it has the path-lifting property To deduce that covering maps have the unique path-lifting propertyKeywords for Lecture 10 Lifting of a map, lifting of a path, lifting property, unique-lifting property, uniqueness-of-lifting property, Covering Homotopy Theorem, local homeomorphism, unique path-lifting property, existence of lifting, fundamental group, universal covering

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