15. MTH 5354/6154 (AUGUST 2021) TOPOLOGY I
This is a graduate level course of Topology for Fall 2021. You may find the course material here.
We will follow our IISER, Pune Ph. D. syllabus (please find the attachment).
Evaluation:
1. Mid-Sem-30%
2. End-Sem-30%.
3. Class tests-10%+10%+10%+10%
Class timing:
Tuesday 4-5 pm
Wednesday 11 am
Thursday 3 pm
Announcements:
There will be 2 tests before mid-sem as part of the continuous evaluation and 2 tests after mid-sem.
Details of Lectures:-
Week 0
1. :-Introduction to the course, quotient topology, different topological spaces.
2. :- Introduction to fundamental groups.
Week 1
3.:-Calculations of Fundamental groups. Assignment 1 uploaded.
4. Statement and some applications of Seifert-Van-Kampen's theorem.
Week 2
5. Proof of the Seifert-Van-Kampen's theorem.
6. Covering spaces. Assignment 2 uploaded.
Class test 1 (02/09/21):
Week 3
7. Path and homotopy lifting theorem, general lifting criteria.
8. Deck transformations, group actions.
9. Groups actions, regular/normal coverings, universal covering spaces.
Assignment 3 uploaded.
Week 4
10. Construction of Universal covering space for semi-locally simply connected spaces.
12. Classification theorem for covering spaces.
13. CW complexes,
Week 5
13. fundamental groups of CW complexes, Covering
spaces of CW complexes.
14. Higher homotopy groups, Commutativity of higher homotopy groups, Higher homotopy groups of covers, (Serre) fibrations,
Differential geometry
Week 1
1. Smooth manifolds.
2. Smooth maps between manifolds.
.
Week 2.
Lie goups, Bump functions
Partition of unity. Extension Lemma, Implicit and inverse function theorem for Euclidean space and Manifolds
Tangent vectors for \R^n.Tangent space for arbitrary manifolds, Push-forwards.
Week .
Tangent bundles.
Introduction to Vector bundles.
Diwali
Week
Submanifold Implicit function theorem Manifolds revisted, Immersions, Submanifolds, submersions.
-Sections, cotangent bundles. Differential Forms.
Assignment 5 uploaded.
Class test 3 on 11/11/21
Week .
Wedge products, Exterior Derivatives.
Vector fields.
Orientation, Riemannian metric, Riemann volume forms.
Week.
Integration on Manifolds using differential forms.
:Definition and basic properties of De Rham cohomology groups. Mayer Vietoris and homotopy properties of De Rham cohomology group.
Class test 4 on 25/11/21
Week.
Stoke's theorem
Level sets, statement of Sard's theorem,
Bundle maps, tensor products, Whitney sums .
Week 16.
:Tutorial
: End semester examination.