Algebraic methods in geometry and topology 1000-135MGT
1. Basic notions of category theory: category, functor, elementary transformations, adjoint functors, Yoneda's lemma, limits and colimits. Additive and abelian categories. Examples from group theory and topology. Groupoids. Presheaves and simplicial sets as examples of functors.
2. Category of modules over a ring as an example of an abelian category. Group ring. Tensor product of modules. Free, projectiva and injective modules, resolutions and generalization to abelian categories.
3. Gradation, filtration and derivation. Chain complexes and homology. Chain homotopy. Derived functors of functors on abelian categories.
4. Derived functors of Hom, tensor products and inverse limits.
Interpretation in terms of extensions. Universal coefficient theorem. Kunneth's formula.
5. Simiplicial complexes and their homology. Nerv of a covering. Cech cohomology of a covering. Soft presheaves and a partition of unity. Cech cohomology of a topological space.
6. Presheaves and sheaves. Direct image and pullback of a sheaf.
Cohomology of sheaves as derived functor of sections. Comparison with Cech cohomology.
7. Locally trivial bundles, vector bundles, principal bundles, covering spaces.
Fundamental group. Classification of bundles in terms of Cech cohomology. The first Stiefel-Whitney formula.
Type of course
Course coordinators
Term 2024Z: | Term 2023Z: |
Learning outcomes
A student should be able to:
- formulate notions from the syllabus and explain them in examples
- formulate theorems from the syllabus and give some chosen proofs
- see categorical nature of various mathematical objects
- illustrate the connections of the sheaf theory and princial bundles with the issues discussed in the framework of other subjects.
Assessment criteria
The final mark will be given on basis of the results of exercises and the final exam. Detailed rules for completing the course are provided in the information on classes in the relevant academic year.
Bibliography
1. Bredon, G. Sheaf Theory. GTM 170. Springer.
2. Bredon, G. Topology and Geometry, Graduate Texts in Mathematics 139, Springer Verlag, New York
1993.
3. Hatcher, A. Algebraic Topology. Cambridge University Press, Cambridge 2002.
4. Fulton, W. Algebraic Topology. A First Course. GTM 153. Springer
5. Gelfand, S.I., Manin, Yu.I. Methods of Homological Algebra. Springer Monographs in Mathematics
2002
6. Husemoller, D. Fibre bundles. Third Edition. GTM 20. Springer.
7. S. Mac Lane, Homology Grundlehren 114, Springer 1963
8. Spanier, E. Algebraic Topology McGraw-Hill
9. Weibel, Ch Homological Algebra
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:
- Bachelor's degree, first cycle programme, Mathematics
- Master's degree, second cycle programme, Mathematics
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: