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401-3571-74L 6 Credits MSC D-MATH

Characteristic Classes

Lecturers & Examiners: Semen Abramyan
VVZ CR n/a

Last Updated: 2026-02-05 16:29:36

Content

– Vector bundles. Operations. – Stiefel-Whitney classes. Applications. – Grassmann Manifolds. Universal bundles. – Principal G-bundles. Universal G-bundles. Milnor construction. – Digression: spectral sequences. Leray-Serre spectral sequence. – Complex vector bundles. Chern classes. – Existence and uniqueness of Stiefel-Whitney and Chern classes. – Cohomology of Grassmann manifolds. Chern character. – Splitting principle. Applications of Chern classes: enumerative geometry. And if time permits: – Pontryagin classes. – Characteristic numbers and bordisms. – Hirzebruch genus. Divisibility of characteristic numbers. – Smooth structures on the 7-sphere. – K-theory. Adams operations. Parallelizability of spheres.

Resources

Literature

[Ad] Adams, John F. Vector Fields on Spheres. Ann. of Math. 75 (1962), 603–632. [At] Atiyah, Michael F. K-theory. N.Y., Benjamin, 1967. [Ha] Hatcher, Allen. Vector Bundles and K-Theory. ( https://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html ) [MS] Milnor, John W.; Stasheff James D. Characteristic Classes. Ann. of Math. Studies, Princeton University Press, Princetion, N. J., 1974. [Mi] Milnor, John W. On Manifolds Homeomorphic to the 7-Sphere. Ann. of Math. (2) 64 (1956), 399—405. [Hi] Hirzebruch, Friedrich. Topological Methods in Algebraic Geometry. Third edition. Springer, Berlin-Heidelberg, 1966.

General Information

Language
English
Levels
MSC

Examination

Type
session examination
Mode
oral 30 minutes
The exam is only offered in the two examination sessions Winter 2025 and Summer 2025.

Course Components

Type Title Time & Place Hours
lecture Characteristic Classes
  • Tue 10:15-12:00 (HG D 5.2)
  • Thu 13:15-14:00 (HG E 1.2)
3 h weekly

Offered In