Characteristic classes of vector bundles and their applications 1000-1M19KCW
1. Real and complex vector bundles. Constructions coming from linear algebra. Pull-back. Structural group of a vector bundle. Orientability. Riemannian metric. Tangent and normal bundles. Canonical bundle.
2. Homotopy classification of vector bundles. Isomorphism of the group of one dimensional bundles with the cohomology group of the projective space (real and complex case).
3. Generalised multiplicative cohomology theories. Leray- Hirsch theorem. Orientability of vector bundles – geometric and cohomological definition and their equivalence. Complex oriented cohomology
4. Axiomatic definition of characteristic classes of vector bundles.
5. Splitting principle and the construction of Stiefel – Whitney and Chern classes. Pontriagin classes.
6. Cohomology operations. Steenrod squares. Computation of Stiefel – Whitney classes by Steenrod squares. Extension of Stiefel – Whitney classes to topological manifolds.
7. Obstruction theory and the interpretation of Stiefel-Whitney and Chern classes in these terms.
8. Information on Chern classes in de Rham cohomology.
9. Application of characteristic classes in solving geometric problems: theorems on embedding manifold into euclidean space, parallelizability of orientable smooth 3 - dimensional closed manifolds.
10. Characteristic numbers and genus of a manifold. Hirzebruch's theorem on signature (information)
Main fields of studies for MISMaP
mathematics
Type of course
Mode
Prerequisites
Prerequisites (description)
Learning outcomes
1. Knowledge of the notion of a vector bundle, basic constructions and homotopy classification of vector bundles.
2. Understanding of the splitting pronciple and the construction of characteristic classes.
3. Ability to interpret characteristic classes as obstructions to existence of sections.
4. Familiarity with Steenrod squares cohomology operations and expressing Stiefel Whitney classes in these terms.
5. Ability to compute characteristic classes of bundle examples and to use these computations to answer questions concerning geometric and topological properties of some smooth manifolds.
Bibliography
Robert R. Bruner, Michael Catanzaro, J. Peter May Characteristic classes. 1974
Ralph L. Cohen The Topology of Fiber Bundles. Lecture Notes, Dept. of Mathematics, Stanford University. 1998
E. Dyer, Cohomology theories, Lecture Note Series, W. A. Benjamin, Inc., New York-Amsterdam, 1969
D. Husemoller Fiber Bundles. Third Edition. Graduate Texts in Mathematics 20. Springer 1993
Ib Madsen Lectures on Characteristic Classes in Algebraic Topology. 1986
John Milnor & James D. Stasheff Characteristic Classes. Annals of Mathematics Studies 76, Princeton University Press.
Robert M. Switzer, Algebraic topology— homotopy and homology. Die Grundlehren der math. Wissenschaften, Band 212, Springer-Verlag, Berlin, 1975
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: