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 the Lecture:

– To understand the notion of homotopy of paths in a topological space

– To understand concatenation of paths in a topological space

– To sketch how the set of fixed-end-point (FEP) homotopy classes of loops at a point

becomes a group under concatenation, called the First Fundamental Group

– To look at examples of fundamental groups of some common topological spaces

– To realise that the fundamental group is an algebraic invariant of topological spaces which

helps in distinguishing non-isomorphic topological spaces

– To realise that a first classification of Riemann surfaces can be done based on their fundamental

groups by appealing to the theory of covering spaces

Keywords:

Path or arc in a topological space, initial or starting point and terminal or ending point of a path,

path as a map, geometric path, parametrisation of a geometric path,

homotopy, continuous deformation of maps, product topology, equivalence of

paths under homotopy, fixed-end-point (FEP) homotopy, concatenation of paths,

constant path, binary operation, associative binary operation, identity element for

a binary operation, inverse of an element under a binary operation, first fundamental group, topological invariant