MA3403 Algebraic Topology I - Fall 2025

Oral exams will take place 2-5 December.

Exam Schedule (now with rooms)

Sample exam questions

I will begin each exam by asking (part of) one of these questions; since you see them in advance, some of them may require a bit more thought than a typical oral exam question.

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.

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).

We will not follow a textbook, but the lectures will mostly follow the notes from the last time I taught this course:

• [H], R. Haugseng, 2020 Notes

For another perspective on the same material (with more pictures), you can also look at Gereon Quick's notes:

• [Q] G. Quick, 2018 Notes

It may also be helpful to consult some of the books listed below - for example, Hatcher's book gives a more geometric view of the subject, which some may prefer to the more algebraic approach in the lectures.

Suggested exercises for 4 September (especially the ones in bold):

• From section 2.2 (categories): 2.4, 2.5
• From section 2.3 (products): 2.6, 2.12, 2.13,
• From section 2.4 (quotients): 2.15 (parts (i) for the circle and (ii))
• From section 2.5 (homotopies): 2.16, 2.17, 2.18
• From section 2.6 (path-components): 2.20

Suggested exercises for 25 September:

• From section 3.8 (disjoint unions): 3.2, 3.3, 3.4
• From section 4.2 (exact sequences): 4.1, 4.2, 4.3 (parts (i) and (ii))
• From section 4.3 (SES of chain complexes): 4.4, 4.5
• From section 4.4 (Eilenberg-Steenrod axioms): 4.7

Suggested exercises for 9 October:

• From section 4.8 (Mayer-Vietoris): Compute the homology of the n-sphere by covering it with two discs (hemispheres) that overlap in a little neighbourhood of the equator; start with the circle and continue by induction. (Exercise 4.10 is a more general version of this.) Also: 4.8, 4.9
• From section 5.1 (Pushouts): 5.1, 5.2, 5.3
• From section 5.2 (Cell complexes): Find a cell structure on the torus, e.g. by thinking of it as a square with opposite sides identified. (What changes if you glue the square differently to get the Klein bottle or projective plane?)

Suggested exercises for 23 October:

• From section 5.5 (Sequential colimits): 5.6, 5.7, 5.9, 5.10
• From section 5.9 (Cellular homology): 5.11, 5.12, 5.13
• From section 6.2 (Chain homotopies): 6.3
• From section 6.4 (Locality and excision): Show that for subsets A,B of X whose interior cover X we have a short exact sequence of chain complexes 0 → S(A∩B) → S(A)⊕S(B) → S'(X) → 0, where S'(X) denotes the small chains with respect to A,B. Use the locality theorem to obtain another construction of the Mayer-Vietoris sequence from this.
• From section 6.5 (Barycentric subdivision): 6.4

Suggested exercises for 6 November:

• From section 7.1 (tensor products): 7.1, 7.2, 7.3
• From section 7.2 (homology with coefficients): 7.7, 7.8, 7.9
• From section 7.3 (Tor): 7.10
• From section 7.4 (universal coefficient theorem): 7.11
• From section 8.1 (Hom): 8.1, 8.2, 8.3, 8.4, 8.5
• From section 8.2 (Singular cohomology): 8.6, 8.7, 8.8, 8.9, 8.10

Suggested exercises for Wednesday 19 November:

• From section 8.5 (Cellular cohomology): 8.11
• From section 9.1 (Tensor products of chain complexes): 9.1, 9.4(i)
• From section 9.2 (Eilenberg-Zilber and Künneth): 9.6
• From section 10.1 (Cross and cup products): 10.1, 10.2
• From section 10.3: 10.5, 10.6 (and 10.7)
• From section 11.3 (cap products): 11.2,
• From section 11.4 (applications to cup products): 11.3, 11.5

• Iver Holten-Mobakk (iverho@ntnu.no)
• David Persson (david.persson@ntnu.no)
• Mathias T. Thorkildsen (mathiatt@stud.ntnu.no)

The following are due to Neil Strickland.

• Torus folding
• Stereographic projection
• The Klein bottle
• R^2 minus a point is homeomorphic to S^1
• RP^1 is homeomorphic to S^1
• Obtaining S^2 by gluing two discs

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.

Some book on category theory:

• [L] T. Leinster, Basic Category Theory, CUP, 2014
• [Ma] S. Mac Lane, Categories for the Working Mathematician, Springer, 1971.
• [R] E. Riehl, Category theory in context, Dover, 2016.