Introduction to Algebraic Topology, Part I (Prof. Anant R. Shastri, IIT Bombay) | Mathematics

Introduction to Algebraic Topology, Part I

Introduction to Algebraic Topology, Part I. Instructor: Prof. Anant R. Shastri, Department of Mathematics, IIT Bombay. This course is central to many areas in modern mathematics. The subject itself saw tremendous growth during 1950 and currently has attained a matured status. The syllabus I have chosen is common to MA5102 at IIT Bombay and AFS-III program of the National Centre for Mathematics. It has enough material common to the syllabi followed by several Universities and IITs in the country and goes beyond. Nevertheless it has a different flavour liked by a variety of students. (from nptel.ac.in)

Introduction

Lecture 01 - Basic Problem in Topology

Lecture 02 - Concept of Homotopy

Lecture 03 - Bird's Eye-view of the Course

Lecture 04 - Path Homotopy

Lecture 05 - Composition of Paths

Lecture 06 - The Fundamental Group

Lecture 07 - Computation of π₁ (S¹)

Lecture 08 - Computation Continued

Lecture 09 - Computation Continued: Some Applications

Lecture 10 - Van-Kampen's Theorem

Lecture 11 - Function Spaces

Lecture 12 - Quotient Maps

Lecture 13 - Group Actions

Lecture 14 - Examples of Group Actions

Lecture 15 - Assorted Results on Quotient Spaces

Lecture 16 - Quotient Constructions Typical to Algebraic Topology

Lecture 17 - Quotient Constructions Continued

Lecture 18 - Relative Homotopy

Lecture 19 - Construction of a Typical SDR

Lecture 20 - Generalized Construction of SDRs

Lecture 21 - A Theoretical Application

Lecture 22 - The Harvest

Lecture 23 - NDR Pairs

Lecture 24 - General Remarks

Lecture 25 - Basics of Affine Geometry

Lecture 26 - Abstract Simplicial Complexes

Lecture 27 - Geometric Realization of Simplicial Complexes

Lecture 28 - Topology on K

Lecture 29 - Simplicial Maps

Lecture 30 - More Examples of Polyhedrons

Lecture 31 - Point Set Topological Aspects

Lecture 32 - Barycentric Subdivision

Lecture 33 - Finer Subdivisions

Lecture 34 - Simplicial Approximation

Lecture 35 - Sperner Lemma

Lecture 36 - Invariance of Domain

Lecture 37 - Proof of Controlled Homotopy

Lecture 38 - Links and Stars

Lecture 39 - Homotopical Aspects of Simplicial Complexes

Lecture 40 - Homotopical Aspects Continued

Lecture 41 - Covering Spaces and Fundamental Groups

Lecture 42 - Lifting Properties

Lecture 43 - Homotopy Lifting

Lecture 44 - Relation with the Fundamental Group

Lecture 45 - Regular Covering

Lecture 46 - Lifting Problem

Lecture 47 - Classification of Coverings

Lecture 48 - Classification Continued

Lecture 49 - Existence of Simply Connected Coverings

Lecture 50 - Toward Construction of Simply Connected Coverings

Lecture 51 - Properties Shared by Total Space and Base

Lecture 52 - Examples

Lecture 53 - G-Coverings

Lecture 54 - Fibred Products and Pull-backs

Lecture 55 - Classification of G-Coverings

Lecture 56 - Proof of Classification

Lecture 57 - Pushouts and Free Products

Lecture 58 - Existence of Free Products

Lecture 59 - Free Products and Free Groups

Lecture 60 - Seifert-Van Kampen Theorems

Lecture 61 - Applications

Lecture 62 - Applications Continued

References

Introduction to Algebraic Topology, Part I
Instructor: Prof. Anant R. Shastri, Department of Mathematics, IIT Bombay. This course is central to many areas in modern mathematics.