# 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 11: Fundamental groups as Fibres of the Universal covering Space

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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 11 To see that the formation of the fundamental group is a covariant functorial operation, from the category whose objects are pointed topological spaces and whose morphisms are base-point-preserving continuous maps, to the category whose objects are groups and whose morphisms are group homomorphisms To deduce the path-lifting property for a covering map as a consequence of the Covering Homotopy Theorem To deduce from the Covering Homotopy Theorem that the fundamental group of a covering space can be identified naturally with a subgroup of the fundamental group of the space being covered To note that the inverse image of a point (fibre over a point) under a covering map may be identified with the space of cosets of the fundamental group (based at a point fixed above) inside the fundamental group at the point below To note that the universal covering of a space may be pictured as a fibration consisting of fundamental groups over that spaceKeywords for Lecture 11 Covering Homotopy Theorem, stationary homotopy, lifting of a homotopy, path-lifting property, category, objects of a category, morphisms of a category, covariant functor, functorial operation, fundamental group as a covariant functor, pointed topological space, group action, transitive action, fundamental group, universal covering, subgroup, cosets of a subgroup in a group

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