Homotopy and the First Fundamental Group

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