This is a basic course in algebraic topology where we introduce the notion of fundamental groups, covering spaces, methods for computing fundamental groups using Seifert Van Kampen theorem and some applications such as the Brouwer’s fixed point theorem, Borsuk Ulam theorem, fundamental theorem of algebra. We discuss some classical groups and their fundamental groups.
The second part of the course concerns singular homology theory and would cover all the standard machinery such as homotopy invariance of homology, relationship with the fundamental group, excision and the Mayer Vietoris sequence. After discussing the relative versions, the course closes with the proof of the famous Jordan Brouwer separation theorem.
Sl No.
Topics
1
Introduction
2
Preliminaries from general topology
3
More Preliminaries from general topology
4
Further preliminaries from general topology
5
Topological groups
6
Test -I
7
Paths, homotopies and the fundamental group
8
Categories and functors
9
Functorial properties of the fundamental group
10
Brouwer’s theorem and its applications
11
Homotopies of maps. Deformation retracts
12-13
Fundamental group of the circle
14
Test -II
15
Covering projections
16
Lifting of paths and homotopies
17
Action of Π1(X, x0) on the fibers p-1(x0)
18
The lifting criterion
19
Deck transformations
20
Orbit spaces
21
Test -III
22
Fundamental groups of SO(3, R) and SO(4, R)
23-24
Coproducts and push-outs
25
Adjunction spaces
26
The Seifert Van Kampen theorem
27
Test -IV
28
Introductory remarks on homology theory
29-30
Singular complex of a topological space
31
The homology groups and theori functoriality
32
Abelianization of the fundamental group
33
Homotopy invariance of homology
34
Small simplicies
35
The Mayer Vietoris sequence
36
Maps of spheres
37
Relative homology
38
Excision theorem
39
Test -V
40
Inductive limits
41
Jordan Brouwer separation theorem
General Topology.
W. S. Massy, A basic course in algebraic topology, Springer Verlag, 1991. Indian reprint, New
Delhi 2007.
J. Vick, Homology theory, Springer Verlag, 1994.
Detailed Bibliography is provided
Important: Please enable javascript in your browser and download Adobe Flash player to view this site
Site Maintained by Web Studio, IIT Madras. Contact Webmaster: nptel@iitm.ac.in
