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The course will give an introduction to Morse Theory. This theory studies the topology of smooth manifolds through real-valued smooth functions whose critical points satisfy a certain non-degeneracy condition. We will investigate how the homotopy type is related to critical points and how the homology of a manifold can be calculated through Morse functions.


Autumn 2011 (Monday, October 10 to Friday, December 16)


  • Tue 11:05 - 11:55


Basic knowledge of Differentiable Manifolds and Algebraic Topology is necessary. This can be obtained through the Core Courses MAGIC063 and MAGIC064.


  • Smooth functions, non-degenerate critical points, Morse functions.
  • Morse Lemma.
  • Morse functions on spheres, projective spaces, orthogonal groups, configuration spaces of linkages.
  • Homotopy type, cell decompositions of manifolds.
  • Existence of Morse functions, cobordisms.
  • gradient flows, stable and unstable manifolds.
  • resonant Morse functions, ordered Morse functions.
  • Morse homology, Morse inequalities.
  • Calculations for projective spaces.
  • Introduction to the h-cobordism theorem.


Dirk Schuetz
Phone (0191) 334 3089
Photo of Dirk Schuetz


Photo of Robin Allan
Robin Allan
Photo of Suliman Alsaeed
Suliman Alsaeed
Photo of Daniel Jones
Daniel Jones
Photo of Benjamin Lang
Benjamin Lang
Photo of Vladimir Lukiyanov
Vladimir Lukiyanov
Photo of Liangang Ma
Liangang Ma
Photo of Simon StJohn-Green
Simon StJohn-Green
Photo of Dragomir Tsonev
Dragomir Tsonev
Photo of Kenneth  Uda
Kenneth Uda
Photo of Marco Wong
Marco Wong


Lectures on the h-cobordism theoremMilnor
An Invitation to Morse TheoryNicolaescu
Morse theoryMilnor
Lectures on Morse homologyBanyaga and Hurtubise
Topology and GeometryBredon
Foundations of differentiable manifolds and Lie groupsWarner


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The Assessment for this course will be via a take-home examination, which will be made available shortly after the end of the course on 13 December. The examination will consists of five questions, and you will need to obtain 40% to pass the course. The deadline for completing the examination is 13 January 2012.

Final Examination

Files:Exam paper
Deadline: Friday 13 January 2012 (1413.7 days ago)

See file for instructions.