Stable Homotopy Theory – University of Copenhagen

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Department of Mathematical Sciences > Research > Topology, Functional Analysis and Algebra > Topology > Stable Homotopy Theory

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Stable Homotopy Theory

taught in Block 2, 2012/13 by Markus Szymik.

Purpose. The aim of this course is to introduce Master and PhD students to the homotopy theory of spectra in the sense of algebraic topology.

Time and Place. The course takes place Mon 13-15 and Wed 10-12. We started on Nov 19, and we will end on Jan 27. The place is 04.4.01.

Exercise sessions for the course are scheduled for Wed 13-16 and Anssi Lahtinen will be responsible for them.

Contents. I intend to cover the Spanier-Whitehead category and duality, spectra and their stable homotopy groups, as well as Adams spectral sequences. While this appears to be a logical order, for various reasons the sequence of lectures will take another course.

  1. Introduction (Exercises)
  2. Spectra and the stable homotopy category (Exercises)
  3. Examples (Exercises)
  4. Steenrod algebra and modules (Exercises)
  5. Examples
  6. The classical Adams spectral sequence (Exercises)
  7. Examples
  8. Methods of computation (Exercises)
  9. Examples
  10. The cobar complex
  11. Examples (Exercises)
  12. The unoriented bordism ring
  13. Realizing morphisms of stable homotopy modules
  14. Realizing stable homotopy modules (Exercises)
  15. The decomposition of stable homotopy
  16. Miscellaneous

Software. We will do a fair amount of computation during the course and during the excercises. It has proved to be instructive to compare the results with those offered by a computer. Bob Bruner's software package, which he has made available online at, is very good for this. We will help everyone with the installation during the course and the exercises.

References.  For the time being, here is a small selection of the texts that I can recommend to accompany the lectures.

J.F. Adams. Stable homotopy and generalised homology. Chicago Lectures in Mathematics. University of Chicago Press, 1974. 

A.K. Bousfield, E.M. Friedlander. Homotopy theory of Γ-spaces, spectra, and bisimplicial sets. Geometric applications of homotopy theory, II, 80–130. Lecture Notes in Math., 658. Springer, Berlin, 1978.

R.B. Bruner. An Adams Spectral Sequence primer. A draft of an introduction to the classical Adams spectral sequence. 2009. (link)

P.G. Goerss. The Adams-Novikov Spectral Sequence and the Homotopy Groups of Spheres. Notes from lectures at IRMA Strasbourg, May 7-11, 2007. (link)

J.P.C. Greenlees. Spectra for commutative algebraists. Interactions between homotopy theory and algebra, 149–173. Contemp. Math., 436. Amer. Math. Soc., Providence, RI, 2007. 

D.C. Ravenel. Complex cobordism and stable homotopy groups of spheres. Pure and Applied Mathematics, 121. Academic Press, 1986.

J. Rognes. Lecture notes on the Adams spectral sequence. 2012. (link)

An even longer list of references for the course may appear here, eventually.