Algebraic Topology and Homological Algebra

Course info

• Class hours: Wednesday 8.30-10 & Thursday 10.15-11.45
• Location: room 002
• Office hours: After the classes or ask in email.
• Lecturer: Ádám Gyenge

Course title: Algebraic Topology and Homological Algebra — ATH
Final grade: Homeworks (50%) + 2 midterms (25% + 25%)
Grading scale: 90/80/70/60/50/40

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. Brief review of the basics of topology
2. Simplicial and singular homology
3. Basic homological algebra (chains and homotopies, categories and functors)
4. Relative homology, excision, Mayer-Vietoris theorem
5. Degree, CW-homology
6. Cohomology, ring structure
7. Modules
8. Künneth formula
9. Orientability, Poincare duality
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