Identity element

Mod-02 Lec-08 Homotopy and the First Fundamental Group

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• published: 14 Jun 2013
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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