Lecture Course Differential Topology, Winter Semester 2017/18
Lectures take place on Tuesday 8-10 in H5 and on Thursday 8-10 in H4. There will be exercise classes Thursdays 10-12 in Geom 142.
- necessary: topology (including fundamental group and covering spaces), basics about manifolds (tangent bundle, flows of vector fields, differential forms)
- recommended: some differential geometry (Riemannian geometry, exponential map)
- also helpful: some algebraic topology (homology and cohomology)
There will be oral exams in the week immediately following the semester, i.e. starting February 5.
In differential topology we study the topology of differentiable manifolds and smooth maps between them. In this course we will start with a discussion of basics such as transversality and degree theory and their applications. We will also discuss vector bundles and tubular neighborhoods. Further topics will depend on audience interest and background knowledge, and may include classification questions for vector bundles, the construction of characteristic classes via Chern-Weil theory, Morse theory and surgery constructions, and the Thom-Pontryagin construction.
The exercise sheets are published here:
Some useful references:
| J. Milnor || Topology from the differentiable viewpoint || University Press of Virginia
| R. Bott, L. Tu || Differential Forms in Algebraic Topology || Springer Verlag
| M. Hirsch || Differential Topology || Springer Verlag
| A. Kosinski || Differential Manifolds || Academic Press
| I. Madsen, J. Tornehave || From calculus to cohomology || Cambridge University Press
| J. Robbin, D. Salamon || Introduction to Differential Topology || Book project