MA3403 Algebraic Topology I - Fall 2022

Schedule Room
Lectures: Monday 12:15-14:00 KJL24
Wednesday 12:15-14:00 B3 Oppredning/gruvedrift (343)
Exam: Oral exams 12th - 15th December Schedule in KJL24
Lecturer
Clover May
Office: 1202 Sentralbygg 2
Email: clover [dot] may [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

Tentative schedule below

Week Lecture Date Topic Notes Exercises
1 1 Mon 22/08 Introduction, review (Substitute Drew Heard) [Q, Lecture 1]
2 Wed 24/08 Review of topological spaces, products and coproducts (Canceled)
2 3 Mon 29/08 Introduction, review, topological spaces, categories [Q, Lecture 1], [H, 2.1-2.2] [H, 2.1]
4 Wed 31/08 More categories, products [H, 2.2-2.3] [H, 2.3, 2.4, 2.5]
3 5 Mon 05/09 Products, coproducts, quotients [H, 2.3-2.4] [H, 2.7, 2.15]
6 Wed 07/09 Group actions, homotopies, path-components, fundamental group [H, 2.5, 2.6, 2.7] [H, 2.16, 2.17, 2.18, 2.20]
4 7 Mon 12/09 Fundamental group - video, chain complexes - video, singular simplices - video, free abelian groups - video [H, 2.7, 3.5, 4.2, 3.1, 3.2]
8 Wed 14/09 Singular chains - video, singular homology - video, functoriality - video [H, 3.2, 3.3, 3.5]
5 9 Mon 19/09 H_0, disjoint unions, H_1 [H, 3.6, 3.7, 3.8] [H, 3.1, 3.2, 3.3, 3.4]
10 Mon 21/09 Abelianization, relative homology, long exact sequence [H, 3.7, 4.1, 4.2] [H, 4.1, 4.2, 4.3]
6 11 Mon 26/09 SES of complexes, reduced homology, Eilenberg-Steenrod axioms [H, 4.1, 4.2, 4.3, 4.4] [H, 4.4, 4.5, 4.6]
12 Wed 28/09 Axioms, quotients and excision, homology of spheres [H, 4.4, 4.5, 4.6] [H, 4.7]
7 13 Mon 03/10 Applications, generator for spheres [H, 4.6, 4.7]
14 Wed 05/10 Mayer–Vietoris, Van Kampen, degree [H, 4.8, 4.9], [Hatcher 2.2] [H, 4.8, 4.9, 4.10 ]
8 15 Mon 10/10 Vector fields on spheres, wedges, computations [H, 4.9], [Hatcher 2.2]
16 Wed 12/10 Computations, van Kampen, pushouts, cell complexes [H, 5.1, 5.2], [Hatcher 1.2] [H, 5.1, 5.2, 5.3]
9 17 Mon 17/10 Cell structures for RPn and CPn, ∆-complexes, simplicial homology, simplicial isomorphic to singular [H, 5.2, 5.3, 5.4], [Hatcher 2.1]
18 Wed 19/10 Consequences of homology isomorphism, simplicial examples, cellular homology, cellular examples, cellular isomorphic to singular [H, 5.4, 5.6, 5.8, 5.9], [Hatcher 2.1, 2.2] [H, 5.11, 5.12, 5.13]
10 19 Mon 24/10 Cellular isomorphic to singular - part 1 video, part 2 - video, homology of CPn - video and RPn - video, chain homotopies - video [H, 5.8, 5.9, 5.10, 6.2], [Hatcher, 2.2] [H, 6.3]
20 Wed 26/10 Homotopy invariance - part 1 video, part 2 - video, excision - video, homology with coefficients - video, tensor products - video [H, 6.1, 6.3, 6.4, 6.5, 7.1], [Hatcher 2.1, 2.2] [H, 6.3, 7.1, 7.2, 7.3, 7.4]
11 21 Mon 31/10 Tensor as a functor, tensor right exact [H, 7.1, 7.2] [H, 7.6, 7.7, 7.8, 7.9]
22 Wed 02/11 Axioms for homology with coeffs, Tor, universal coefficient theorem [H, 7.2, 7.3, 7.4, 7.5] [H, 7.10]
12 23 Mon 07/11 More Tor, UCT, Hom functor, singular cohomology [H, 7.3, 7.4, 8.1, 8.2] [H, 7.11, 8.1, 8.2, 8.3, 8.4, 8.5]
24 Wed 09/11 Singular cohomology, cellular and simplicial cohomology, Eilenberg–Steenrod axioms for cohomology, Ext [H, 8.2, 8.3, 8.5], [Hatcher 3.1] [H, 8.6, 8.7, 8.8, 8.9, 8.10]
13 25 Mon 14/11 Ext, UCT for cohomology, cup products in cohomology [H, 8.3, 8.4], [Quick, Lecture 20], [Hatcher, 3.2] [H, 8.11]
26 Wed 16/11 Functoriality and graded-commutativity of ring structure, ring structure for cohomology of spheres, torus, and projective spaces [Quick, Lectures 20 and 21], [Hatcher, 3.2]
14 27 Mon 21/11 Künneth for rings, relative cup products, manifolds, local and global orientation [Quick, Lectures 21 and 22], [Hatcher, 3.2, 3.3]
28 Wed 23/11 Orientations, Kronecker pairing, Poincaré duality and applications [Hatcher 3.3], [Quick, Lectures 22 and 24]


Here is a link to an exercise sheet from a previous version of this course. Here are the solutions.

Link to an article about the Klein bottle. See the first animation to get the Klein bottle from the gluing diagram.

Link to an article about deformation retracts and strong deformation retracts, with nice animations and the example of the topologist's comb.

The proof of the Jordan Curve Theorem can be found in Hatcher 2.B.

Link to a video animating the formation of the genus two torus from its gluing diagram.

Link to all the exercises from Rune Haugseng's notes.

Discourse

We have a Discourse page where you can ask questions about the course, discuss exercises, etc. There is a new user guide for Discourse if you have never used it before: https://meta.discourse.org/t/discourse-new-user-guide/96331 and the NTNU help page is https://wiki.math.ntnu.no/drift/help/forum.

Reference group

The members of the reference group are Denis Bondarenko Bergmann, Eivind Xu Djurhuus, and Isak Drage. Please feel free to pass any feedback on to them.

Course material

We will largely follow portions of the previous lecture notes:

References

Other good lecture notes:

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.
2023-08-16, Clover May