Algebraic methods in geometry and topology

elective courses

Fundamental notions of the category theory, additive and abelian categories. Tensor product in the category of modules. Projective and injective modules, resolvents. Graded groups, chain complexes and their homologies.

Derived functors of Hom and of the tensor product. Presheaves, sheaves and their cohomologies.

Simplicial cohomologies and Cech cohomologies. Coverings and principal bundles; cohomological interpretation.

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.

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

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.

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.