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Summer Research Fellowship Programme of India's Science Academies

Topology and its applications

Arnab Roy

Ramakrishna Mission Residential College (Autonomous), Narendrapur, Kolkata, 700103

Dr. Vamsi Pritham Pingali

Indian Institute of Science, Bengaluru, Karnataka, 560012

Abstract

Studying "shapes" of different objects is the main goal of Topology. But it is not always that easy to distnguish objects (their shapes) using standard methods of point-set topology. In Algebraic topology, we attack this problem through the techniques of algebra. Study of Algebraic Topology needs quite a good amount of group theory, and of course basic point-set topology as prerequisite. In this project, we shall firstly start from some crucial concepts of point-set topology, namely Tychonoff Theorem, a little bit of complete metric spaces, where our main attempt will be generalized, 'Arzela-Ascoli's theorem' and Baire Spaces. Then we shall move on to Algebraic Topology, strictly speaking first homotopy classes and Fundamental group, a very important algebraic tool to measure "holes" in a topological space. Computation of fundamental groups of some topological objects (like- S1S^1, figure-eight) using the methods like, deformation retracts to simpler spaces, covering spaces and using Seifert-Van Kampen theorem will also be studied. In due course, we would also study several applications of fundamental group in proving various results in mathematics. In particular we shall go through the topological proofs of the fundamental theorem of algebra, the Brouwer fixed-point theorem for the disc, the Perron-Frobenius theorem on matrices and the Abel-Ruffini theorem.

Keywords: group, baire Space, Abel-Ruffini theorem, complete metric spaces, fundamental groups, homotopy.

Arnab Roy

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