4-dimensional Manifolds

In this panorama-style lecture course, I shall cover the following topics:

  1. Introduction - ICM 1986
  2. 2-dimensional manifolds - uniformisation and moduli
  3. 3-dimensional manifolds - Perelman's theorem
  4. High-dimensional manifolds - Poincaré's conjecture and surgery theory
  5. Homotopy theory associated with 4-manifolds
  6. Construction of 4-manifolds - Kirby calculus
  7. Topological 4-manifolds - Casson handles and Freedman's topological surgery theory
  8. Geometric structures - complex and symplectic manifolds
  9. Kodaira's classification scheme for complex 2-dimensional manifolds
  10. Instantons and Donaldson's invariants
  11. Monopoles and Seiberg-Witten theory
  12. Stable cohomotopy invariants
  13. Floer theory
  14. Open problems

This panorama-style lecture course covers 14 topics related to the geometry and topology of 4-dimensional manifolds. Each week's lectures will constitute a self-contained introduction to main ideas, concepts and results of the respective area. The lecture course intends to exhibit the interplay between geometry, topology, algebraic geometry, analysis and theoretical physics in this area of research. There are no compulsory prerequisites other than curiosity about and interest in the subject.


  • S. Bauer: Refined Seiberg-Witten invariants
  • S. Donaldson, P. Kronheimer: The geometry of four-manifolds
  • R. Kirby: The topology of 4-manifolds
  • A. Scorpan: The wild world of 4-manifolds