MA3403 Algebraic Topology - Fall 2020
| Schedule | Room | ||
|---|---|---|---|
| Lectures: | Tuesday | 14:15-16:00 | H1 |
| Wednesday | 12:15-14:00 | G21 | |
| Exam: | Oral exams | 30 Nov-2 Dec (online) | |
| Lecturer | |||
| Rune Haugseng | |||
| Office: | 1250 Sentralbygg 2 | ||
| Email: | rune [dot] haugseng [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
The first lecture will be on 18 August.
| Lecture | Date | Topic | Notes | Exercises |
|---|---|---|---|---|
| 1 | 18/08 | Introduction, review | 1.1-2.2 | 1.(1,2), 2.(1,2,3) |
| 2 | 19/08 | Products, quotients, homotopies | 2.3-2.5 | 2.(7,15,17,18) |
| 3 | 25/08 | Path-components, singular chains | 2.6, 3.1-2 | 2.20 |
| 4 | 26/08 | Singular homology, chain complexes | 3.3-3.5 | |
| 5 | 01/09 | H0 and π0, H1 and π1 | 3.6-3.7 | 3.1 |
| 6 | 02/09 | End of π1, homology of disjoint unions, relative homology | 3.7-8, 4.1 | 3.(2,3,4) |
| 7 | 08/09 | Exact sequences | 4.2-3 | 4.(1,2,3,4,5) |
| 8 | 09/09 | Eilenberg-Steenrod axioms, excision and quotients | 4.4-5 | 4.7 |
| 9 | 15/09 | Homology of spheres, applications, Mayer-Vietoris | 4.6-8 | |
| 10 | 16/09 | Examples of Mayer-Vietoris, degrees | 4.8-9 | 4.(8,9,10) |
| 11 | 22/09 | Pushouts, cell complexes | 5.1-2 | 5.(1,2,3) |
| 12 | 23/09 | Delta-complexes, simplicial homology | 5.3-4 | 5.(4,5) |
| 13 | 29/09 | Sequential colimits, homology of cell complexes, simplicial and singular homology | 5.5-7 | 5.(6,7,10) |
| 14 | 30/09 | Cellular homology, homology of real projective space | 5.8-10 | 5.(11,12,13) |
| 15 | 06/10 | Exterior products | 6.1 | 6.(1,2) |
| 16 | 07/10 | Chain homotopies, homotopy invariance, excision, locality | 6.(2-4) | 6.(3,4) |
| 17 | 13/10 | Barycentric subdivision, tensor products | 6.5-7.1 | 7.(1,2,4) |
| 18 | 14/10 | Homology with coefficients, Tor | 7.2-3 | 7.(8,9,10) |
| 19 | 20/10 | Universal coefficient theorem, cellular homology with coefficients, Hom | 7.4-5, 8.1 | 7.11, 8.(1,2,4,5) |
| 20 | 21/10 | (No lecture) | ||
| 21 | 27/10 | Singular cohomology, Ext, universal coefficient theorem for cohomology | 8.2-4 | 8.(6,8,9,10) |
| 22 | 28/10 | Cellular cohomology, tensor products of chain complexes | 8.5,9.1 | 8.11,9.(1,2) |
| 23 | 03/11 | Eilenberg-Zilber and Künneth theorems, cup and cross products | 9.2-3,10.1 | 9.(5,6),10.1 |
| 24 | 04/11 | Ring structures on cochains and cohomology | 10.1-3 | 10.(2,3,4,5,6) |
| 25 | 10/11 | Manifolds | 11.1 | |
| 26 | 11/11 | Orientations, cap products | 11.2-3 | 11.(1,2) |
| 27 | 17/11 | Applications to cup products, cohomology with compact support, proof of Poincaré duality | 11.4-6 | 11.(3,4,5,6) |
| 28 | 18/11 | Review |
Course material
References
We will not follow any particular textbook. The lecture notes from the past two years will give you a good idea of the content of the course:
- [19], 2019 Notes
- [Q] G. Quick, 2018 Notes
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.