2018/2019

## Research Seminar "Algebraic Topology"

Type:
Optional course (faculty)

Delivered by:
Faculty of Mathematics

When:
3, 4 module

Instructors:
Alexei Gorinov

Language:
English

### Course Syllabus

#### Abstract

Everybody knows that the UN declared 2019 the year of the periodic table of elements. Mathematics without Algebraic Topology is like Chemistry without the Periodic Table. We will learn the basic results of Algebraic (alias homotopy) Topology.

#### Learning Objectives

- 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.

#### Expected Learning Outcomes

- 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

#### Course Contents

- 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.

#### Bibliography

#### Recommended Core Bibliography

- Allen Hatcher. (2002). Algebraic topology. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.87FE219C

#### Recommended Additional Bibliography

- 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