Research Seminar "Algebraic Topology"
- The students will gain an understanding of the seminal works in 20-th Century Topology. For example, they will be able to determine the lower bounds on the dimension of real affine spaces where they can embed real projective spaces.
- The students know basic examples of topological spaces
- The students learn to apply Eckmann-Hilton duality
- The students can compute fundamental groups
- The students can compute singular cohomology
- The students can compare homology and homotopy groups
- The students learn Eilenberg-MacLane spaces
- The students learn to compute cohomology groups of fibrations
- The students learn to compute characteristic classes
- Topological spaces: examples, properties, operations.We describe some basic examples of topological spaces and their properties.
- The notion of homotopy. Fibrations and cofibrations. Cone and homotopy fibre. Suspension and loop spaces. Eckmann–Hilton duality.We define the basic notions of algebraic topology.
- Homotopy groups. Exact sequence of a fibration. Coverings and the fundamental group.We describe the geometric meaning of the fundamental group.
- Axiomatics of cohomology theories. Uniqueness theorem. Construction of singular cohomology.We classify various cohomology theories.
- Homology and cohomology of CW-complexes. Hurevich theorem.We describe the relation between homotopy and homology groups.
- Eilenberg-Maclane spaces. Postnikov tower.We describe the Eilenberg-Maclane spaces.
- Leray spectral sequence of a fibration.We explain how to compute the cohomology of the total space of a fibration.
- Characteristic classes of vector bundles: Stiefel – Whitney, Chern, Pontriagin.We define and compute characteristic classes and describe their applications.
- Allen Hatcher. (2002). Algebraic topology. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.87FE219C
- Published Xx Xxxember Xx, Allen Hatcher, Karen Vogtmann, & Nathalie Wahl. (2006). Algebraic & Geometric Topology Volume X (20XX) 1–XXX. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.96F4FAE