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 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 space

Keywords 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