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

Introduction to Algebraic Topology, Part II

Introduction to Algebraic Topology, Part II. Instructor: Prof. Anant R. Shastri, Department of Mathematics, IIT Bombay. As stated above, this is a PG level course in Mathematics, which requires basic knowledge of Linear algebra, Point set topology, and group theory. 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 National Centre for Mathematics. It has enough material common to the syllabus followed by several Universities and IITs in the country and goes beyond. Nevertheless it has a different flavour liked by a variety of students. I have published a book in which one-third of the content is roughly the present course. This book is followed by several universities abroad also for their course. (from nptel.ac.in)

Introduction

Lecture 01 - Introduction

Lecture 02 - Attaching Cells

Lecture 03 - Subcomplexes and Examples

Lecture 04 - More Examples

Lecture 05 - Topological Properties

Lecture 06 - Coinduced Topology

Lecture 07 - Compactly Generated Topology on Products

Lecture 08 - Product of Cell Complexes

Lecture 09 - Product of Cell Complexes Continued

Lecture 10 - Partition of Unity on CW-Complexes

Lecture 11 - Partition of Unity Continued

Lecture 12 - Homotopical Aspects

Lecture 13 - Homotopical Aspects Continued

Lecture 14 - Cellular Maps

Lecture 15 - Cellular Maps Continued

Lecture 16 - Homotopy Exact Sequence of a Pair

Lecture 17 - Categories: Definitions and Examples

Lecture 18 - More Examples

Lecture 19 - Functors

Lecture 20 - Equivalence of Functors Continued

Lecture 21 - Universal Objects

Lecture 22 - Basic Homological Algebra

Lecture 23 - Diagram-Chasing

Lecture 24 - Homology of Chain Complexes

Lecture 25 - Euler Characteristics

Lecture 26 - Singular Homology Groups

Lecture 27 - Basic Properties of Singular Homology

Lecture 28 - Excision

Lecture 29 - Examples of Excision-Mayer Vietoris

Lecture 30 - Applications

Lecture 31 - Applications Continued

Lecture 32 - The Singular Simplicial Homology

Lecture 33 - Simplicial Homology

Lecture 34 - Simplicial Homology Continued

Lecture 35 - CW-Homology and Cellular Singular Homology

Lecture 36 - Construction of CW-chain Complex

Lecture 37 - CW Structure and CW Homology of Lens Spaces

Lecture 38 - Assorted Topics

Lecture 39 - Some Applications of Homology

Lecture 40 - Applications of LFT

Lecture 41 - Jordan-Brouwer

Lecture 42 - Proof of Lemmas

Lecture 43 - Relation between π₁ and H₁

Lecture 44 - All Postponed Proofs

Lecture 45 - Proofs Continued

Lecture 46 - Definitions and Examples

Lecture 47 - Paracompactness

Lecture 48 - Manifolds with Boundary

Lecture 49 - Embeddings and Homotopical Aspects

Lecture 50 - Homotopical Aspects Continued

Lecture 51 - Classification of 1-Manifolds

Lecture 52 - Classification of 1-Manifolds Continued

Lecture 53 - Triangulation of Manifolds

Lecture 54 - Pseudo-Manifolds

Lecture 55 - One Result due to Poincare and Another due to Munkres

Lecture 56 - Some General Remarks

Lecture 57 - Classification of Compact Surface

Lecture 58 - Final Reduction-Completion of the Proof

Lecture 59 - Proof using Fundamental Group

Lecture 60 - Orientability

References

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