Algebraic Topology and Homological Algebra

Course info

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

For BME students

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

For BSM students

Course title: Algebraic Topology — AGT
Final grade: Homeworks + 2 short midterms

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

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)