Course on Algebraic Topology (Fall 2014)

This is a course jointly taught by Ieke Moerdijk and Javier J. Gutiérrez within the Dutch Master's Degree Programme in Mathematics (Mastermath).
The teaching assistant for this course is Joost Nuiten.

  • Place and time: Wednesdays 10:15-13:00 (first session: 17 September). Utrecht University room MIN211 (Minnaert building).
  • Prerequisites: A course in point-set topology and group theory. The intensive course Categories and Modules (9 and 10 September).
  • Test yourself exam: Wednesday, 19 November (2 hours) followed by an exercise session for sheet 9. The solutions for the test yourself exam will be discussed after Lecture 10.
  • Final exam: Wednesday, 21 January, 13:30-16:30. Utrecht University, Educatorium Theatronzaal.
  • Retake exam: There will be a retake exam in April or May (date still to be fixed). Depending on the number of students, it will be a written exam or an oral exam. If you want to participate, please send an email, before April 1, to the secretary of the mathematics department in Nijmegen, Greta Oliemeulen-Löw (
    Unfortunately, we are unable to allow students who already passed the exam to participate in the retake in order to improve the mark.
  • If you are interested in learning more algebraic topology, we are running a course on Topological K-Theory. You can find the details here.

Course description >

The course will start with a reminder about the fundamental group which most students will have seen, and the homotopy relation on maps. Next, we will define the higher homotopy groups and prove some basic properties about them. We will discuss the action of the fundamental group on these groups and and the Serre long exact sequence of a fibration. This will enable us to compute some elementary examples.

We will discuss CW-complexes and a proof of Whitehead's theorem about the construction of maps from the information about homotopy groups. Further, we will use CW-complexes to construct the Postnikov tower of a space, a context in which the famous Eilenberg-Mac Lane spaces come up. We also hope to discuss the Freudenthal suspension theorem, a key result that lies at the basis of stable homotopy theory. If time allows, we will end the course with a discussion of Quillen's axioms for "homotopical algebra", axioms which play a dominant role in much of modern algebraic topology.

Lecture notes >

  • Lecture 1: Homotopy of maps and the fundamental groupoid. (17/09)
  • Lecture 2: Spaces of maps, loop spaces and reduced suspension. (24/09)
  • Lecture 4: Relative homotopy groups and the action of the fundamental group. (08/10)
  • Lecture 5: Fibrations and homotopy fibers. (15/10)
  • Lecture 10: CW approximation and Whitehead's theorem. (26/11) Discussion of test yourself exam.
  • Lecture 11: Postnikov and Whitehead towers. (03/12) (extendedd version, uploaded 09/12)
  • Lecture 12: Further properties of cofibrations. (10/12) (pages 7-10 of the notes for Lecture 8 under this link)
  • Lecture 13: Representable functors and Brown representability theorem. (17/12) (corrected version)

Exercise sheets >

  • Sheet 8 (The exercise session for sheet 8 will take place on 12/11)

References >

Although some books on algebraic topology focus on homology, most of them offer a good introduction to the homotopy groups of a space as well. You can get a good impression of the subject, for example, from the following references:

  • M. Arkowitz, Introduction to homotopy theory. Universitext. Springer, 2011.
  • J. Davis and P. Kirk, Lecture notes in algebraic topology. GTM 35, American Mathematical Society, 2001.
  • A. Hatcher, Algebraic topology. Cambridge University Press, Cambridge, 2002.
  • E. H. Spanier, Algebraic topology. Springer-Verlag, 1981.
  • R. M. Switzer, Algebraic topology - homotopy and homology. Springer-Verlag, 1975.

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