This is a graduate level course of Topology for August 2023. 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:

Mon, We, Fri:12-1 

Announcements:

There will be 2 tests before mid-sem as part of the continuous evaluation and 2 tests after mid-sem.

Class test 1-25/08/23

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. 

4. Statement and some applications of Seifert-Van-Kampen's theorem.

Week 2

5. Covering spaces. 

6.  Path  and homotopy lifting theorem, general lifting criteria.

Week 3

7.  Deck transformations,  group actions.

8.   Groups actions, regular/normal coverings, universal covering spaces.

Class test 1.

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.

Week 

 Submanifold Implicit function theorem Manifolds revisted,  Immersions, Submanifolds, submersions. 

-Sections, cotangent bundles. Differential Forms.

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. 

Week.

 Stoke's theorem

 Level sets, statement of Sard's theorem,

Bundle maps, tensor products, Whitney sums .

 Week 16.

:Tutorial

: End semester examination.