This is a graduate level course on topology.
Please visit the following website for
syllabus of the course.
You will find relevant information about the course.
1. Mid-Sem-30%
2. End-Sem-30%.
3. Presentations-10%+10% (one before midsem and one befor end sem)
5. Short tests-10%+10% (One in end of January and one in end of March).
Details of Lectures:-
We will learn about homology groups this January
this is a tentative list. We may change topics depending on the progress of the class.
Week 1
1. Lecture 1 (03/ 01/2025):-Introduction to Homology groups
Week 2
2. Lecture 2 (06/01/2025):- Simplices, simplicial complexes and Delta Complexes.
3. Lecture 3 (08/01/2025):-Simplicial and Singular Homology.
4. Lecture 4 (10/01/2025) Relation between Homology and Fundamental group.
Week 3
Homotopy Invariance
Relative Homology, Exact sequences, Brouwer’s fixed point for n-disks
Excision
Week 4.
Mayer-Vietoris, Euler Characteristic.
Week 5
CW-complexes, Cellular homology, Homology with coefficients, Statement of Equivalence of simplicial and singular homology
We will learn about Orientation this February.
Week 6
Degree, Axioms for Homology,
We will learn about cohomology groups this March
Week 7 (3/3)
Idea of Cohomology
Cohomology of spaces
Universal Coefficients Theorem
Week 8 (10/3)
Cross and Cup products
Cohomology Ring,
Statement of the basic Kunneth Formula,
Week 9 (17/3)
• Orientation of topological manifolds, Orientation in terms of Homology
Statement of Poincare Duality
Week 10 (24/3)
Computation of Cohomology rings, Statement of DeRham comparison theorem
Week 11
Class test 2 on 31st March 2025
Statement of Lefschetz fixed point theorem, Statement of Invariance of domain,
Week 12
Problem solving
Week 13
Presentation.