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:.
Problem Set 1
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:
| Apr 08 | introductory remarks; definition of topological manifolds, Grassmannians as example, smooth atlases and smooth structures, smooth maps between smooth manifolds, submersions, immersions and embeddings, Whitney embedding theorem (first version)
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| Apr 22 | sets of measure zero, behaviour under smooth maps, easy Whitney embedding theorem with comments, regular and critical points and values, statement and discussion of Sard's theorem, outline of proof of smooth version (following Milnor's book); you can find Whitney's paper on his example here
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| Apr 29 | manifolds with boundary, examples: sublevel sets of regular values, preimages in manifolds with boundary of regular values, smooth Brouwer fixed point theorem, Brouwer's fixed point theorem for continuous maps; transversality of a map to a submanifold, first consequences
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| May 6 | transversality in the presence of boundary, transversality for families, transverse approximation relative to a closed subset
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| May 6 | mapping degree mod 2: homotopy invariance, first examples and consequences, orientations and integer mapping degree, homotopy invariance, examples
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| May 13 | linking number for a pair of knots; vector bundles: definitions, first examples, morphisms and trivializations, non-uniqueness of trivializations (when they exist), orientations, constructions with vector bundles: direct sum, tensor product etc, pullback bundles
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