Global homotopy theory

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

Stefan Schwede, schwede (at) math.uni-bonn.de

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.

Topics

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.

Lecture videos

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.

References:

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