Advanced Topics in Topology (V5D1)
Equivariant stable homotopy theory
Winter term 2026/27

Lecture course, Tuesdays, 10:15-12:00 and Thursdays, 14:15-16:00 in SR 1.008
Stefan Schwede

Topics

The class is an introduction to equivariant stable homotopy theory. We will develop the general theory for compact Lie groups of equivariance, with some emphasis on finite groups; we will use orthogonal G-spectra as our model. Some topics to be covered include: equivariant stable homotopy groups, the `genuine' G-equivariant stable homotopy category, the Wirthmüller isomorphism, transfers, genuine and geometric fixed points, and the tom Dieck splitting. Along the way, we'll discuss many examples.

References:
- A. Blumberg, The Burnside category. Lecture notes for M392C (Topics in Algebraic Topology), Spring 2017, U Texas, Austin.
- M. Mandell, J. P. May, Equivariant orthogonal spectra and S-modules. Mem. Amer. Math. Soc. 159 (2002), no. 755, x+108 pp.
- S. Schwede, Lecture notes on equivariant stable homotopy theory.
- Chapter 3 of S. Schwede, Global homotopy theory, New Mathematical Monographs 34. Cambridge University Press, Cambridge, 2018. xviii+828 pp. [download]
- The first lectures will begin with some unstable equivariant homotopy theory. This part of the lecture notes were typset by Alessandro Nanto, but potential mistakes, if any, are mine.

Survey articles:
- J. F. Adams, Prerequisites (on equivariant stable homotopy) for Carlsson's lecture. Algebraic topology, Aarhus 1982, 483-532. Lecture Notes in Math. 1051, Springer-Verlag, 1984.
- J. P. C. Greenlees, J. P. May, Equivariant stable homotopy theory. Handbook of algebraic topology, 277-323. North-Holland, Amsterdam, 1995. - S. Schwede, Universal symmetries: Global equivariant homotopy theory. Expository article, to appear in the proceedings of the 9th ECM, Seville.

Prerequisites

Exam

There will be oral exams after the end of the class.