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Homotopy and the First Fundamental Group
>> YOUR LINK HERE: ___ http://youtube.com/watch?v=ebnmXIQrB4Q
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/... • 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
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