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Seminar "Morse Theory" (Winter Semester 2023/24)

The introduction and assignment of talks will take place on Wednesday, July 26, 2023 at 11:30 pm in seminar room -1.008 in the basement of the mathematics building (20.30).

Seminar: Thursday 14:00-15:30 20.30 SR 2.066
Lecturer Prof. Dr. Wilderich Tuschmann
Office hours: by appointment
Room 1.002 Kollegiengebäude Mathematik (20.30)
Email: tuschmann@kit.edu
Office hours:
Room Kollegiengebäude Mathematik (20.30)

Morse theory studies the topology of finite- or infinite-dimensional differentiable manifolds by means of investigating the extremal behavior of smooth functions defined on these manifolds. It has diverse and far-reaching mathematical and physical applications, for example, to the geodesic structure of Riemannian manifolds, Milnor's exotic spheres, Smale's h-cobordism theorem and the solution of the generalized Poincaré conjecture, or Bott's periodicity theorems for the unitary and orthogonal groups.

Based on John Milnor's textbook Morse Theory and further literature, the foundations and classical applications of Morse theory will be worked out in the seminar.

Aufbau eines 2-Torus


  • Morse lemma, sublevel sets and attaching cells
  • Morse inequalities
  • Morse functions
  • The index theorem
  • Approximation and topology of path space
  • Lie groups and symmetric spaces
  • Path spaces and homotopy groups
  • The Bott periodicity theorem for U(n)


The seminar is intended for students with sound knowledge of differentiable manifolds and differential geometry. Topological foundations about homotopy groups, homology theory and CW-complexes will be covered in the seminar.


W. M. BOOTHBY, An introduction to differentiable manifolds and Riemannian geometry. 2nd edition. Pure and Applied Mathematics, 120. Academic Press, Inc., Orlando, FL, (1986).

A. HATCHER, Algebraic topology. Cambridge University Press, Cambridge, (2002).(online verfügbar unter http://www.math.cornell.edu/~hatcher/#ATI)

J. MILNOR, Morse theory. Annals of Mathematics Studies, No. 51 Princeton University Press, Princeton, N.J. (1963).
(Eine neu geTexte Fassung findet sich unter https://oldbookstonew.blogspot.com/)

R. STÖCKER & H. ZIESCHANG, Algebraische Topologie. Eine Einführung. 2nd edition. Mathematische Leitfäden. B. G. Teubner, Stuttgart, (1994).

R. M. SWITZER, Algebraic Topology – Homotopy and Homology. Die Grundlehren der mathematischen Wissenschaften, Band 212. Springer-Verlag, New York-Heidelberg, (1975).