NFMV039 Algebraic topology Spring 25
This page contains the program of the course. Other information, such as learning outcomes, teachers, literature and examination, are in a separate course PM. You can register for the course here.
Program
The schedule of the course will be in TimeEdit.
Lectures
There will be three lectures/week throughout reading period 4. The lectures are at 15:15-17:00 on Mondays, 10:00-11:45 on Tuesdays, and 15:15-17:00 on Wednesdays. All lectures are in MVF23. There are no lectures during the Easter week and the week after. The only exception to the above is that the lecture on the Wednesday 30th of April is from 08:00-9:45. This is due to the 30th of April being a half-day.
| Day | Sections | Content |
|---|---|---|
|
Lec 1 24/3 |
1.1, 1.3 | Intro to the course, review of fundamental group, covering spaces etc. |
|
Lec 2 25/3 |
0 | Cell/CW-complexes, fundamental examples of topological spaces, operations on topological spaces |
|
Lec 3 26/3 |
0 | Continuation of topics in lecture 2 |
|
Lec 4 31/3 |
Homological algebra, short exact sequences, long exact sequence in homology | |
|
Lec 5 1/4 |
2.1 | Singular/simplicial homology |
|
Lec 6 2/4 |
2.1 |
Singular/simplicial homology |
|
Lec 7 7/4 |
2.1 |
Excision |
|
Lec 8 8/4 |
2.2 |
Mayer-Vietoris |
|
Lec 9 9/4 |
2.2 |
Mayer-Vietoris and examples of computations |
|
Lec 10 28/4 |
2.2 |
More examples of computations, Euler characteristic |
|
Lec 11 29/4 |
2.2 |
Homology with coefficients |
|
Lec 12 30/4 |
2.2 |
Equivalence of simplicial and singular homology for Δ-complex; Cellular homology |
|
Lec 13 5/5 |
3.1 |
Cohomology |
|
Lec 14 6/5 |
3.1 |
Universal coefficients theorem |
|
Lec 15 7/5 |
3.2 |
Cup product and the cohomology ring |
|
Lec 16 12/5 |
3.2 |
Cup product and the cohomology ring |
|
Lec 17 13/5 |
3.2, 3.B |
Künneth formula |
|
Lec 18 14/5 |
3.3 |
Poincaré duality |
|
Lec 19 19/5 |
3.3 |
Poincaré duality |
|
Lec 20 20/5 |
4.1 |
Higher homotopy groups |
|
Lec 21 21/5 |
4.1-3 |
Higher homotopy groups, long exact sequence associated to fibrations, Freudenthal suspension theorem |
|
Lec 22 26/5 |
4.1 |
Whitehead's theorem |
|
Lec 23 27/5 |
Additional topic: other cohomology theories |
|
|
Lec 24 28/5 |
Additional topic: other cohomology theories |
Coursework exercises
Around the lecture on the 9th of April you will be given a piece of coursework to hand-in, with deadline around the time the lectures resume again after Easter. This will be a few exercises that you need to write up the solutions to. This is a compulsory part of the evaluation of the course.
Link to the homework: Homework.pdf
Recommended exercises
| Week | Exercises |
|---|---|
| 1 | 0.1-5, 0.11, 0.14, 1.1.12, 1.1.14, 1.3.1-3,1.3.8-9 |
| 2 | 2.1.14-15, 2.1.30-31, 2.1.1-3, 2.1.7, 2.1.11 |
| 3 | 2.1.16-17, 2.1.22, 2.1.27, 2.2.2-4, 2.2.9-10, |
| 4 | 2.2.14-15, 2.2.21-23, 2.2.27, 2.2.41 |
| 5 | 3.1.1-3, 3.1.6, 3.1.9, 3.2.1-3 |
| 6 | 3.2.11, 3.3.6, 3.3.11 |
| 7 | 4.1.1-2, 4.1.12-13, 4.1.15, 4.2.32 |
| 8 |
Reference literature:
- Algebraic topology, Allen Hatcher. Our core reference for the course, all chapter number refers to this book. Available as e-book on Hatcher's webpage.
- Ext(A,B): online lecture on Ext groups by Richard E. Borcherds. https://www.youtube.com/watch?v=fNHr1CMyuvI