Lecture Differential Topology

Textbooks:

1. "An Introduction to manifolds", Loring W. Tu
2. "Differential Topology", Morris W. Hirsch

Further used literature: 

1. "Differential forms in Algebraic Topology", Bott--Tu
2. "Topology from the differentiable view point", Milnor
3. Notes on differential forms part 6: Top cohomology, Poincare duality and degree, Texas Austin
4. Math 703 Part 2: Vector bundles, Weimin Chen 

Course task: In this course we learn about manifolds, i.e. locally Euclidean spaces, and how we transfer the anlysis, e.g. integration and differentiation, from R^n to smooth manifolds. The relation between manifolds and Euclidean spaces is like the relation between special relativity and Newton mechanics, i.e. global to local. Manifolds show up in many places, as for example as phase-spaces and energy surfaces (mechanics), as parametrized hypersufaces, space-time (physics), Lie groups, indifference surfaces (economics).  .

Course content:

1. Manifolds, submanifolds, tangent bundle, differential maps, immersions, submersions, approximation,
2. Whitney embedding theorem
3. Regular value theorem and transversality
4. Morse-Sard theorem
5. Stoke's  theorem
6. Differential forms and the de Rham complex
7. Mayer-Vietoris sequence
8. Vector bundles

Room and Time: S506 (IMS), Tuesdays and Fridays 15:00-16:40.

Problem sheets and lecture notes:

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

Lecture notes:  week1 week2 week3 week4 week5 week6 week7 week8

week9 week10 week11 week12 week13 week14-1 week14-2 week15-1 week15-2

week16

Problem sheets: Sheet1 Sheet2 Sheet3 Sheet4 Sheet5 Sheet6  Sheet7 Sheet8

Sheet9 Sheet10 Sheet11 Sheet12 Sheet13 Sheet14