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Characteristic classes of vector bundles and their applications

General data

Course ID: 1000-1M19KCW
Erasmus code / ISCED: 11.1 Kod klasyfikacyjny przedmiotu składa się z trzech do pięciu cyfr, przy czym trzy pierwsze oznaczają klasyfikację dziedziny wg. Listy kodów dziedzin obowiązującej w programie Socrates/Erasmus, czwarta (dotąd na ogół 0) – ewentualne uszczegółowienie informacji o dyscyplinie, piąta – stopień zaawansowania przedmiotu ustalony na podstawie roku studiów, dla którego przedmiot jest przeznaczony. / (0541) Mathematics The ISCED (International Standard Classification of Education) code has been designed by UNESCO.
Course title: Characteristic classes of vector bundles and their applications
Name in Polish: Klasy charakterystyczne wiązek wektorowych i ich zastosowania
Organizational unit: Faculty of Mathematics, Informatics, and Mechanics
Course groups: (in Polish) Przedmioty 4EU+ (z oferty jednostek dydaktycznych)
(in Polish) Przedmioty obieralne na studiach drugiego stopnia na kierunku bioinformatyka
Elective courses for 2nd stage studies in Mathematics
ECTS credit allocation (and other scores): (not available) Basic information on ECTS credits allocation principles:
  • the annual hourly workload of the student’s work required to achieve the expected learning outcomes for a given stage is 1500-1800h, corresponding to 60 ECTS;
  • the student’s weekly hourly workload is 45 h;
  • 1 ECTS point corresponds to 25-30 hours of student work needed to achieve the assumed learning outcomes;
  • weekly student workload necessary to achieve the assumed learning outcomes allows to obtain 1.5 ECTS;
  • work required to pass the course, which has been assigned 3 ECTS, constitutes 10% of the semester student load.

view allocation of credits
Language: English
Main fields of studies for MISMaP:

mathematics
physics

Type of course:

elective monographs

Prerequisites:

Algebraic topology 1000-135TA

Prerequisites (description):

Basic properties of singular (co-)homology or de Rham cohomology.

Mode:

Blended learning

Short description:

Vector bundles and their homotopy classification. The axiomatic definition of characteristic classes; existence proved using the splitting principle and in case of real bundles also the Steenrod squares. Characteristic classes as obstructions to the existence of sections. Applications of characteristic classes to geometry of smooth manifolds: their immersions in euclidean space and parallelizability. Characteristic numbers, bordism of smooth manifolds.

Full description:

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)

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

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.

This course is not currently offered.
Course descriptions are protected by copyright.
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