The Importance of the Path-lifting Property

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 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 property

Keywords 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