Global homotopy theory

Lecture course, University of Bonn, Summer term 2020
Advanced Topics in Topology (V5D1)

Stefan Schwede, schwede (at)

This class was taught during the summer term 2020, in the midst of the Covid-19 pandemic; it consists of 22 video lectures that are available on this youtube channel; individual youtube and download links can also be found below.


This class is an introduction to global homotopy theory, i.e., those phenomena in equivariant homotopy theory where all groups act simultaneously and compatibly. Here 'all groups' refers to compact Lie groups (or to a specific class of compact Lie groups). Examples of such global equivariant phenomena are equivariant cohomotopy, equivariant K-theory, equivariant bordism or Borel cohomology theories.

We use orthogonal spectra under global equivalences as our model, the main reference is the book Global homotopy theory. Some topics include: orthogonal spectra as models for global homotopy types; global classifying spaces; global Mackey functors; the global model structure; the global stable homotopy category. Along the way, we'll discuss many examples, including suspension spectra of global spaces, the global Thom spectra mO and MO, and symmetric product spectra.

Prerequisites for this class are a solid knowledge of basic algebraic topology and (non-equivariant) homotopy theory; some familiarity with equivariant homotopy theory for a fixed group is helpful, but I'll give a brief review during the first lectures. I discuss the prerequisites in more detail at the beginning of the first lecture.

Lecture videos

Lecture 1: The setup   (youtube download)
Lecture 2: Orthogonal spectra as models for global stable homotopy types   (youtube download)
Lecture 3: Transfers   (youtube download)
Lecture 4: The double coset formula   (youtube download)
Lecture 5: Global classifying spaces   (youtube download)
Lecture 6: Equivariant homotopy groups of global suspension spectra   (youtube download)
Lecture 7: Global functors   (youtube download)
Lecture 8: The global Thom spectrum mO   (youtube download)
Lecture 9: Equivariant homotopy groups of mO   (youtube download)
Lecture 10: The global Thom spectrum MO   (youtube download)
Lecture 11: The global model structure   (youtube download)
Lecture 12: The triangulated global homotopy category   (youtube download)
Lecture 13: Global versus G-equivariant stable homotopy   (youtube download)
Lecture 14: Left and right induced global homotopy types   (youtube download)
Lecture 15: Generating t-structures by compact objects   (youtube download)
Lecture 16: The standard t-structure on the global stable homotopy category   (youtube download)
Lecture 17: Global equivariant properties of symmetric products   (youtube download)
Lecture 18: G-equivariant homotopy groups of symmetric products   (youtube download)
Lecture 19: Splitting global functors at orthogonal, unitary and symplectic groups   (youtube download)
Lecture 20: Stable splittings of global classifying spaces and regularity of equivariant Euler classes   (youtube download)
Lecture 21: Stable global splittings of Stiefel manifolds   (youtube download)
Lecture 22: Stable splittings via linear algebra   (youtube download)

As bonus material, you might want to view the video of my talk "Universal properties of global equivariant Thom spectra" at BIRS (February 15, 2016), which complements the material of lectures 8 and 9; or the video of my talk "The global stable homotopy category" at BIRS (June 23, 2016), which complements the material of lectures 12--14.


The main reference is:

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

Some useful references with background on equivariant stable homotopy theory are:
- 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.
- 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, 27.08.2020