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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:

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)
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
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


 
  Seitenanfang  Impressum 2025-04-29, Janko Latschev