Differential Topology Spring 2025

Main idea of the course:

Manifolds are the underlying objects in many subjects like Mathematics, Physics,

Economics. the easiest way to see this is to consider the fiber of a regular value of a smooth map. The point is that those objects are locally n-spaces, but in general not globally. Differential topology is designed to provide tools to answer global questions about smooth manifolds. 

Topics: 

1. Smooth manifolds (If time allows, then we talk about approximation) 
2. Submanifolds
3. Differential forms and the de Rham complex
4. De Rham cohomology and de Rham's Theorem
5. Stokes' Theorem
6. Vector bundles 
7. Mayer-Vietoris sequence

Literature:  There are notes from my course in Spring 2023, but the syllabus will change a bit, since we will not emphisize on approximation theory. 

1. Morris W. Hirsch,  Differential Topology
2. Loring W. Tu, An introduction to Manifolds
3. Raoul Bott and Loring W. Tu, Differential Forms in Algebraic Topology

Room and Time: IMS S507: Tuesdays(odd weeks) and Thursdays (all weeks) 

3pm-4:40 pm. Starting on Tuesday, February the 18th. 2025. 

Grade: 40% homework and 60% final

Problem sheets: 

There will be given a problem sheets every week which can be found  here as  well as the lecture notes.

Sheet 1 Sheet 2 Sheet 3 Sheet 4 Sheet 5 Sheet 6 Sheet 7 Sheet 8 Sheet 9 

Sheet 10-11 Sheet 12 Sheet 13 Sheet 14-15

The homework needs to be handed in on Thursdays before the  

 lecture. Late submissions do not count. 

Office hour: Even Tuesdays 3pm-4:40pm

Lecture notes: The syllabus can be found here.

Week 1 de Rham cplx of R^n, Def of manifold and RP^n 

             [BottTu, Ch1.S1], [Tu, Ch2.(S5 and S7)]

Week 2 Submanifolds, smooth maps and extrinsic def of TM

             [Hirsch, Ch1.(S1 and 2)]

Week 3 Intrinsic description for TM. Embedding of a compact manifold into R^q

            [Tu, Ch3 (S8)], [Hirsch, Ch1(S3)]

Week 4 Manifolds with boundary

            [Hirsch, Ch1 (S4)]