Algebraic Topology and Homological Algebra

Course info

• Class location and hours: Tuesday 14.30-16 (room H306) & Thursday 14.30-16 (room H405A).
• Location: BME campus, building H
• Office hours: Before or after the classes (13.30-14.15 and 16-16.30), or ask in email.

Schedule

• Lectures start on 11/02 Tue
• 18/04-25/04 is Easter Break at BME
• We will finish by 08/05
• There will be some make-up classes

For BME students

Neptun code: BMETEAGMsMATHA-00
Credits: 5
Final grade: Homeworks + 2 short midterms

For BSM students

Course title: Algebraic Topology and Homological Algebra — ATH
Final grade: Homeworks + 2 short midterms

Extra classes (tentatively): occasionally Monday 8-10 at BSM site. BME students are also welcome.

• 03/03: exercise solving
• 24/03: Knot theory
• two more in April
• …

Course description

The goal of the course is to provide an introduction to the basic notions of homology and cohomology theory, and show some simple (and some more sophisticated) applications of these techniques in topology and algebra. Ideas from homology are present in all modern directions of mathematics, and we will show several appearances of those as well.

Planned topics:

1. Simplicial and singular homology
2. Basic homological algebra (chains and homotopies, categories and functors)
3. Degree, CW-homology
4. Cohomology, ring structure
5. Orientability, Poincare duality
6. Obstruction theory
7. Fiber bundles, principal bundles
8. Classification of vector bundles
9. Characteristic classes
10. Advanced homological algebra, Hom, Tensor
11. Derived functors

Prerequisites:

Basic algebra: vector spaces, groups, factor groups, homomorphisms, rings and ideals.
Basic topology: topological spaces, continuous maps, homeomorphisms, homotopy, constructions.

Here is a summary/review of basic topology (from a course at the Queen Mary University, London).

Text:

• Allen Hatcher: Algebraic Topology
• Joseph Rotman: Introduction to Homological Algebra

Homework

• HW 1 Due: 25/02
• HW 2
• HW 3
• HW 4
• HW 5

Midterms

1. March 18

2. May 6

Format: 30 minutes, 2 questions (1 theory + 1 problem very similar to a homework exercise)