MA3403 Algebraic Topology - Fall 2021
Schedule | Room | ||
---|---|---|---|
Lectures: | Monday | 10:15-12:00 | MA24 |
Wednesday | 14:15-16:00 | H1 | |
Exam: | Oral exams | Monday 13th - Wednesday 15th December | |
Lecturer | |||
Drew Heard | |||
Office: | 1258 Sentralbygg 2 | ||
Email: | drew [dot] k [dot] heard [at] ntnu [dot] no |
What this course is about
Studying geometric objects by associating algebraic invariants to them is a powerful idea that has influenced many areas of mathematics. For example, determining the existence of a map between spaces (often a difficult task) may be translated into deciding whether an algebraic equation has a solution (sometimes a piece of cake). One of the birthplaces of this idea is Algebraic Topology. The goal of the course is to introduce some of the most important examples of such invariants, namely singular homology and cohomology groups. Along the way we are going to calculate many examples and see some applications.
See the study handbook for more information.
What you need to know before this course
The key idea of algebraic topology is to apply algebra to study topological spaces, so to follow the course you should have some background in both algebra and topology. Ideally this means you have already taken the courses "Introduction to topology" (TMA4190) and "Rings and modules" (MA3201); the course "Differential topology" (TMA4192) also covers much of the required background in topology. That said, if you are motivated it is still possible to follow the course even if you haven't taken these other courses yet, as the amount of material that we will actually use is fairly limited. In this case it is a good idea to gain some familiarity with topological spaces, for example by looking at the books by Jänich [J] or Munkres [Mu2].
In addition, it is recommended to take the course "Homological algebra" (MA3204) this semester (but it is not required as we will cover the small amount of homological algebra we need).
If you have any questions, feel free to contact me!
Lecture Plan
Lecture | Date | Topic | Notes | Exercises | Video |
---|---|---|---|---|---|
1 | 23/08 | Introduction, review | [Q, Lecture 1] | ||
2 | 25/08 | Review of topological spaces, products and coproducts | [H, 2.1-2.4] | 2.1,2.3,2.7 | |
3 | 30/08 | Homotopies, path-components, singular chains | [H,2.5-2.7,3.1] | 2.16, 2.17,2.20 | |
4 | 01/09 | Singular chains, singular homology | [H,3.1- 3.4] | Link | |
5 | 06/09 | Functoriality and H_0 | [H,3.6 - 3.8], [Q, Lecture 4] | 3.2,3.3,3.4. | |
6 | 08/09 | Relative homology and exact sequences | [H,4,1.4,2], [Q,Lecture 5] | 4.1-4.3 | Link |
7 | 13/09 | SES of complexes, Eilenberg-Steenrod axioms | [H,4.3, 4.4] | 4.4,4.5 | |
8 | 15/09 | Quotients and excision, homology of spheres | [H,4.5,4.6] | 4.7 | Success! |
9 | 20/09 | Homology of spheres, applications, Mayer-Vietoris | [H,4.6-4.8] | Try and compute the homology of the sphere using Mayer–Vietoris! | |
10 | 22/09 | Example with Mayer–Vietoris, degree | [H,4.8-4.9] | 4.8,4.9,4.10 | Link (unfortunately, sound only in the second half) |
11 | 27/09 | Degree, cell complexes | [H,4.9-5.2] | 5.1,5.2,5.3 | Lecture 11 |
12 | 29/09 | Cell complexes, sequential colimits | [H,5.2,5.5] | 5.6-5.10 | Lecture 12 |
13 | 04/10 | Cellular chains | [H,5.5,5.6,5.8] | Lecture 13 | |
14 | 07/10 | Cellular differential, homology of RPn* | [H,5.9,5.10] | 5.(11,12,13) | Lecture 14 |
15 | 11/10 | Exterior products | [H,6.1] | 6.1,6.2 | Lecture 15 |
16 | 13/10 | Chain homotopies, homotopy invariance, excision, locality | [H,6.2-.4] | 6.3,6.4 | Lecture 16 |
17 | 18/10 | The locality theorem | [H,6.5] | Lecture 17 | |
18 | 20/10 | Tensor products, homology with coefficients | [H,7.1,7.2] | 7.(1,2,4) | Lecture 18 |
19 | 25/10 | Homology with coefficients, Tor | [H,7.2,7.3] | 7.(8,9,10) | Lecture 19 |
20 | 27/10 | The universal coefficient theorem | [H,7.4,7.5] | 7.11 | Lecture 20 Unfortunately both microphones stopped working in the second half, sorry! |
21 | 01/11 | Hom functor, singular cohomology | [H,8.1-8.2] | 8.(1,2,4,5,6) | Lecture 21** |
22 | 03/11 | Singular cohomology, the UCT for cohomology | [H,8.3-8.5] | 8.(8,10) | Lecture 22 |
23 | 08/11 | Cup products in cohomology | [Quick, Lecture 20, Lecture 21] | Lecture 23 | |
24 | 10/11 | Applications of cup products | [Quick, Lecture 21] | Lecture 24 | |
25 | 15/11 | Manifolds, orientations, cap products | [Hatcher, Section 3.3] | Lecture 25 | |
26 | 16/11 | Applications to cup products | [Hatcher, Section 3.3] | Lecture 26 | |
27 | 22/11 | Question session | |||
28 | 23/11 | Review/exam prep | Review notes | Review |
Here is a link to an exercise sheet from a previous version of this course. Here are the solutions.
Here is the image of the Klein bottle used on 22/09.
* Note there is an error both in my lecture and in the lecture notes: The degree of the antipodal map on S^n is(-1)^{n+1} because it is represented by the negative of the identity matrix, but we apply it to S^{n-1}, so the degree is (-1)^n.
** Small correction to the lecture: the cochain complex computing the cohomology of a point should have maps 0,id,0,id,0,id, etc.
Reference group
The reference group consists of Even Aslaksen, Bjørnar Hem, and Chileshe Mwamba. We will meet for the first time on Wednesday 15/09. Please pass any feedback on to them.
Course material
We will not follow any particular textbook. The previous lecture notes will give you a good idea of the content of the course:
- [H], R. Haugseng, 2020 Notes
- [Q] G. Quick, 2018 Notes
References
Other good lecture notes:
- [Mi] H. Miller, Lectures on Algebraic Topology I
- [G] M. Groth, Algebraic Topology I and II
Some interesting books:
- [H] A. Hatcher, Algebraic Topology, Cambridge University Press, 2000.
- [Mu] J.R. Munkres, Elements of Algebraic Topology, Westview Press, 1996.
- [F] W. Fulton, Algebraic Topology - A First Course, Springer-Verlag, 1995.
- [V] J.W. Vick, Homology Theory - An Introduction to Algebraic Topology, Springer, 1994.
- [Ma] J.P. May, A Concise Course in Algebraic Topology, Chicago Lectures in Mathematics, 1999.
- [MS] J. Milnor, J. Stasheff, Characteristic Classes, Princeton University Press, 1974.
Some books on general topology:
- [J] K. Jänich, Topology, Springer, 1984.
- [Mu2] J.R. Munkres, Topology: a first course, Prentice-Hall, 1975.