# PhD course: Equivariant stable homotopy theory

**Time: **
Mon 2023-03-13 15.00 - 17.00

**Location: **
Cramér room, Albano

**Participating: **
Stefan Schwede (University of Bonn, visiting professor at SU)

**Content**

The class is an introduction to equivariant stable homotopy theory (for finite groups), including an outlook towards globally-equivariant phenomena. 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. I will discuss the orthogonal spectrum model for global spectra, including the structure of global homotopy groups ("global Mackey functors") and the global stable homotopy category. Along the way, we'll discuss examples and relate the models to the higher categorical perspective.

Prerequisites: solid general knowledge of algebraic topology and basic homotopy theory. Ideally, students would already be acquinted with non-equivariant stable homotopy theory and some unstable equivariant homotopy theory (G-CW-complexes, Elmendorf's theorem). I will occasionally offer a higher categorical perspective on things; for that, some familiarity with infty-categories is helpful.

Schedule: Mondays 15-17, starting March 13. (Ten lectures.)

Examination: PhD students who would like to get credits for the course can do an oral exam.

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. - S Schwede, Global homotopy theory, New Mathematical Monographs 34. Cambridge University Press, Cambridge, 2018. xviii+828 pp.

- S Schwede, Lectures on Global Homotopy Theory, Videos hosted by YouTube https://www.youtube.com/channel/UCYvnwhhcZoYMu0iSNvm7bRQ/videos 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.