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 www.nptel.iitm.ac.in/syllabus/...
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

[@inverse_functor]

This is the best one explaining homotopy.

[@mittaltushant]

Such a clear explanation of the fundamental group in just 50 minutes ! I love how you leave out certain (intuitive enough) things as exercises so that a decent amount of material is covered. Thanks a lot, Professor Balaji

[@bnouadam]

thank u. very accesible. well structrured mind reflected on easy demo

[@anowarali668]

thanks