# MA3403 Algebraic Topology I - Fall 2023

Schedule Room
Lectures: Tuesday Wednesday 12:15-14:00 14:15-16:00 R60 Berg B21 KJL22
Friday 12:15-14:00 R D4-132
Exam: Oral exams 4th-11th December schedule in KJL22
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.

## 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 Tue 22/08 Introduction, review [Quick Lecture 1]
2 Fri 25/08 Review topological spaces, categories [H 2.1, 2.2] [H 2.1]
2 3 Tue 29/08 Universal properties - video, products - video, coproducts - video, quotients - video [H 2.2, 2.3, 2.4] [H 2.3, 2.4, 2.5, 2.7]
4 Fri 01/09 Quotient examples - video, projective spaces - video, group actions - video, homotopies and equivalences - video, retracts - video, path components - video [H 2.4, 2.5, 2.6] [H 2.15, 2.16, 2.17, 2.18, 2.20]
3 5 Mon 04/09 Fundamental group - video, chain complexes - video [H 2.7, 3.5, 4.2]
6 Fri 08/09 Singular simplices - video, free abelian groups - video, singular chains - video, singular homology - video [H 3.1, 3.2, 3.3]
4 7 Wed 13/09 Functoriality - video 1), natural isomorphisms - video, H_0 - video, disjoint unions - video [H 3.5, 3.6, 3.8] [H 3.1, 3.2, 3.3, 3.4]
8 Fri 15/09 H_1, abelianization [H 3.7], [Hatcher 2.A]
5 9 Wed 20/09 Relative homology - video, relative long exact sequence - video, connecting homomorphism and exactness - video [H 4.1, 4.2, 4.3, 4.4] [H 4.1, 4.2, 4.3]
10 Fri 22/09 Reduced homology - video, extension problems - video, Eilenberg-Steenrod axioms - video, excision and quotients - video [H 4.4, 4.5] [H 4.7]
6 11 Wed 27/09 Homology of spheres, applications, generator for spheres [H 4.6, 4.7]
12 Fri 29/09 More on generator, Mayer–Vietoris, van Kampen [H 4.7, 4.8], [Hatcher 2.2] [H 4.8, 4.9, 4.10 ]
7 13 Wed 04/10 Mayer–Vietoris examples, degree, vector fields on spheres [H 4.8, 4.9], [Hatcher 2.2]
14 Fri 06/10 More vector fields, wedges, computations [H 4.9], [Hatcher 2.2]
8 15 Wed 11/10 Computations, pushouts, cell complexes [H 5.1, 5.2] [H 5.1, 5.2, 5.3]
16 Fri 13/10 Cell structures for RPn and CPn, ∆-complexes, simplicial homology [H 5.2, 5.3, 5.4], [Hatcher 2.1]
9 17 Wed 18/10 Simplicial isomorphic to singular, consequences of homology isomorphism, simplicial examples, cellular homology [H 5.4, 5.6, 5.8, 5.9], [Hatcher 2.1, 2.2]
18 Fri 20/10 Cellular homology, cellular examples, cellular isomorphic to singular [H 5.8, 5.9], [Hatcher 2.2] [H, 5.11, 5.12, 5.13]
10 19 Wed 25/10 Cellular isomorphic to singular, homology of CPn and RPn [H 5.10], [Hatcher 2.1, 2.2]
20 Fri 27/10 Chain homotopies, homotopy invariance, excision, homology with coefficients [H 6.1, 6.2, 6.3, 6.4, 6.5], [Hatcher 2.1, 2.2] [H 6.3]
11 21 Wed 01/11 Homology with coefficients, tensor products, tensor as a functor [H 7.1, 7.2], [Hatcher 2.2] [H 7.1, 7.2, 7.3, 7.4, 7.6]
22 Fri 03/11 Tensor right exact, axioms for homology with coeffs [H 7.1, 7.2, 7.5] [H 7.7, 7.8, 7.9]
12 23 Wed 08/11 Tor, universal coefficient theorem, Hom functor, singular cohomology [H 7.3, 7.4, 7.5, 8.1, 8.2], [Hatcher 3.1] [H 7.10, 7.11]
24 Fri 10/11 Singular, cellular, and simplicial cohomology [H 8.2, 8.5], [Hatcher 3.2] [H 8.1, 8.2, 8.3, 8.4, 8.5, 8.6, 8.7]
13 25 Wed 15/11 Eilenberg–Steenrod axioms for cohomology, Ext, UCT for cohomology [H 8.2, 8.3, 8.4], [Hatcher 3.2] [H 8.8, 8.9, 8.10, 8.11]
26 Fri 17/11 Cup products in cohomology, functoriality and graded-commutativity of ring structure, ring structure for cohomology of spheres [Quick Lecture 20], [Hatcher 3.2]
14 27 Wed 22/11 Cohomology ring of the torus - video, Künneth for rings - video, cohomology of wedges - video, manifolds - video, orientable manifolds - video [Quick Lectures 21 and 22], [Hatcher 3.2, 3.3]
28 Fri 24/11 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. More exercises and solutions can be found at the course page.

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.

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.

Link to an introduction to persistent homology in topological data analysis by Otter, Porter, Tillmann, Grindrod, and Harrington.

Link to a user's guide to topological data analysis by Munch.

## Reference group

The members of the reference group are Jakob Lanser, Michele Mignani, and Peter Rotar. Please feel free to pass any feedback on to them.

## Course material

We will largely follow portions of the previous lecture notes:

and the book:

## References

Other good lecture notes:

Some other interesting books:

• [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.
1)
There is an error in the induced map around 22:00. The composition induces a map from H_n(C) to H_n(D). It does not necessarily factor through Z_n(D).