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.
Evaluation:-
1. Mid-Sem-30%
2. End-Sem-30%.
3. Assignments-20%
4. Presentations-10%
5. Short tests-10%
Assignments will be uploaded every Friday in
this website. Please submit the assignment
by next Thursday.
I will upload the lecture notes in this website.
Details of Lectures:-
Week 1
1. Lecture 1 (03/ 08/2015):-Introduction to Surfaces.
2. Lecture 2 (05/08/2015):- Introduction to fundamental groups.
3. Lecture 3 (06/08/2015):-Calculations of Fundamental groups.
(07/08/2015):- Assignment 1 uploaded.
Week 2
4. Lecture 4 (10/08/2015):- Statement and some applications of Seifert-Van-Kampen's theorem.
5. Lecture 5 (11/08/2015):- Proof of the Seifert-Van-Kampen's theorem.
I was on academic leave from 12/08/2015-23/08/2015 visiting IMSC, Chennai.
(14/08/2015):- Assignment 2 uploaded.
Week 3
6. Lecture 6. (13/08/2015):- By Tathagata Mandal. Covering spaces.
7. Lecture 7. (17/08/2015):-By Tathagata Mandal. Path homotopy and homotopy lifting theorem.
8. Lecture 8. (19/08/2015):-By Tathagata Mandal. Revision of proof of Seifert-Van-Kampen's theorem.
(21/08/2015):- Assignment 3 uploaded.
Week 4
9. Lecture 10 (24/08/2015)- Introduction to covering spaces, Monodromy theorem, Injectivity
theorem and statement of general Lifting theorem.
10. Lecture 10 (26/08/2015)- General lifting theorem, Universal covering space.
11. Lecture 11(27/08/2015)- Motivation/Construction of Universal covering space for semi-locally
simply connected spaces.
(28/08/2015):- Assignment 4 uploaded.
Week 5
12. Lecture 12 (31/08/2015):- Classification theorem for covering spaces. Construction of Universal covering
(leftovers from previous class).
13. Lecture 13 (02/09/2015):-Deck transformations, Galois/normal coverings, group actions.
14. Lecture 14 (03/09/2015):-CW complexes, fundamental groups of CW complexes, Covering
spaces of CW complexes.
(03/09/2015) Assignment 5 uploaded.
Presentations by students on 05/09/2015. Topic presented by the students:
a. Fundamental group of circle.
b. Fundamental theorem of algebra.
c. Borsuk-Ulam theorem.
Week 6
Short test on 07/09/2015.
15. Lecture 15 (08/09/2015):- Smooth manifolds.
16. Lecture 16 (09/09/2015): Smooth maps between manifolds.
17. Lecture 17 (10/09/2015):-Bump functions and partition of unity.
Week 7
I was on academic leave this week. I was attending a conference on "Iwasawa theory and "p-adic Hodge theory" at the University of Bielefeld, Germany.
18. Lecture 18 (14/09/2015):- Implicit and inverse function theorem for Euclidean space and Manifolds. Lecture by Tathagata Mandal.
19. Lecture 19 (16/09/2015):-Tangent vectors for \R^n. Lecture by Tathagata Mandal.
Week 8
Mid-sem Week. No lecture.
Week 9
Mathematics Symposium at the IISER, Pune. No Lecture.
Week 10.
20. Lecture 20 (05/10/2015) Tangent space for arbitrary manifolds, Push-forwards.
21. Lecture 21 (07/10/2015) Tangent bundles.
22. Lecture 22 (08/10/2015) Vector fields.
10/10/2015:Assignment 6 uploaded.
Week 11.
23. Lecture 23 (12/10/2015) Introduction to Vector bundles.
24. Lecture 24 (14/10/2015) Bundle maps, Whitney sums.
25. Lecture 25 (15/10/2015):-Sections, cotangent bundles, tensor products.
16/10/2015:Assignment 7 uploaded.
Week 12.
26. Lecture 26 (19/10/2015): Differential Forms,
No class on 21/10/2015 and 22/10/2015: Festival holiday.
Week 13.
27. Lecture 27 (26/10/2015): Wedge products, Exterior Derivatives.
28. Lecture 28 (28/10/2015): Orientation, Riemannian metric, Riemann volume forms.
29. Lecture 29 (29/ 10/2015): Integration on Manifolds using differential forms.
(30/10/2015): Assignment 8 uploaded.
30. Lecture 30 (31/10/2015): Presentation by students: Topics presented:
1. Fundamental group of topological manifold countable.
2. Frobenius theorem.
3. Left invariant vector fields and Lie algebras.
Week 14.
31. Lecture 31 (02/11/2015): Definition and basic properties of De Rham cohomology groups.
32. Lecture 32 (03/11/2015):-Mayer Vietoris and homotopy properties of De Rham cohomology
group. The class will be at 10. 30 am (unusual timing).
No lecture on 04/11/2015 as I am visiting ISI, Kolkata.
07/11/2015: Assignment 9 uploaded.
Week 15.
(09/11/2015): Exams,
33. Lecture 33 (10/11/2015): Immersions, Submanifolds, submersions.
No class on 11/11/2015. Diwali Holiday.
34. Lecture 34 (12/11/2015): Level sets, statement of Sard's theorem.
13/11/2015: Assignment 10 uploaded.
Week 16.
35. Lecture 35 (16/11/2015):Revision, Stoke's theorem.
36. Lecture 36 (18/11/2015):Higher homotopy groups, commutativity of higher homotopy groups, Higher homotopy groups of covers, Serre fibrations.
No lecture on 19/11/2015.
End-semester Exam on 25/11/2015.