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Differential Topology

Module MA5122

This Module is offered by TUM Department of Mathematics.

This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.

Module version of WS 2012/3 (current)

There are historic module descriptions of this module. A module description is valid until replaced by a newer one.

Whether the module’s courses are offered during a specific semester is listed in the section Courses, Learning and Teaching Methods and Literature below.

available module versions
WS 2012/3SS 2012

Basic Information

MA5122 is a semester module in English language at Master’s level which is offered irregular.

This module description is valid from SS 2012 to WS 2018/9.

Total workloadContact hoursCredits (ECTS)
150 h 45 h 5 CP

Content, Learning Outcome and Preconditions


This course will provide an introduction to basic concepts of differential topology. We will discuss immersions, submersions and embeddings, critical points and Sard's theorem, Whitney's embedding theorem, as well as some mapping degree theory. Applications include fixed point theorems and the Borsuk-Ulam theorem.

Learning Outcome

After successful completion of the module, students are able to analyze topological problems from a differentiable viewpoint and to see differential problems from a topological perspective. They master in particular the use of Sard's theorem, Brouwer's fixed point theorem and the Borsuk-Ulam theorem. Moreover, they know and understand the concepts of embeddings and immersions.


MA1001 Analysis 1, MA1002 Analysis 2, MA1101 Linear Algebra and Discrete Structures 1, MA1102 Linear Algebra and Discrete Structures 2.
Helpful but not essential: MA2004 Vector Analysis.

Courses, Learning and Teaching Methods and Literature

Courses and Schedule

VO 2 Differential Topology Wolf, M. Mon, 14:15–15:45, MI 00.09.022
UE 1 Differential Topology (Exercise Session) Wolf, M.

Learning and Teaching Methods

The module is offered as lectures with accompanying practice sessions. In the lectures, the contents will be presented in a talk with demonstrative examples, as well as through discussion with the students. The lectures should animate the students to carry out their own analysis of the themes presented and to independently study the relevant literature. Corresponding to each lecture, practice sessions will be offered, in which exercise sheets and solutions will be available. In this way, students can deepen their understanding of the methods and concepts taught in the lectures and independently check their progress. At the beginning of the module, the practice sessions will be offered under guidance, but during the term the sessions will become more independent, and intensify learning individually as well as in small groups.




- V. Guillemin, A. Pollack, Differential Topology.
- J.W. Milnor, Topology from the differentiable viewpoint.

Module Exam

Description of exams and course work

The exam will be in written (60 minutes) or oral (25 minutes) form, depending on the number of participants. Students demonstrate that they have gained deeper knowledge of definitions and main mathematical tools and results in differential topology. The students are expected to be able to derive the methods, to explain their properties, and to apply them to specific examples.

Exam Repetition

The exam may be repeated at the end of the semester.

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