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 7: Moebius Transformations Make up Fundamental Groups of Riemann Surfaces

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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 the Lecture - Every good topological space possesses a unique simply connected covering space called the Universal covering space- The fundamental group of the topological space shows up as a subgroup of automorphisms of its universal covering space- The universal covering map expresses the target space as the quotient of the universal covering space of the target, by the fundamental group of the target- A covering map can be used to transport Riemann surface structures from source to target and vice-versa, thus making it into a holomorphic covering map - Any Riemann surface is the quotient of the complex plane, or the upper half-plane, or the Riemann sphere by a suitable group of Moebius transformations isomorphic to the fundamental group of the Riemann surface - The study of any Riemann surface boils down to studying suitable subgroups of Moebius transformations Keywords Covering map, covering space, admissible open set or admissible neighborhood, simply connected covering or universal covering, local homeomorphism, Riemann surface structure inherited by a topological covering of a Riemann surface, uniformisation, fundamental group, Moebius transformation

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