Janko Latschev
Lecture Course Differential Topology, Summer Semester 2025
In differential topology we study the topology of differentiable manifolds and smooth maps between them. This course aims to bridge (some of) the gap between what is usually taught about these topics in a first course in differential geometry and what is often assumed in more advanced courses.
We will start with a discussion of basics such as transversality and degree theory and their applications. We will also discuss tubular neighborhoods for submanifolds and some of their uses. Further topics will depend on audience interest and background knowledge.
Prerequisites: - necessary: topology (including fundamental group and covering spaces), basics about manifolds (definitions, implicit and inverse function theorems, 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 no exercise classes, but I encourage students to work on exercise sheets which are posted semi-regularly 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 |
Log of the lectures:
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