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Janko Latschev

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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: