Global homotopy theory / Lecture 6: Equivariant homotopy groups of global suspension spectra

Lecture 6 of "Global homotopy theory": I explain how the 0-th equivariant stable homotopy groups of the suspension spectrum of an orthogonal space can be described in terms of unstable homotopy information. When applied to the global classifying space of a compact Lie group, this completes the calculation of the morphism groups in the global Burnside category.

Contents:
00:11 Review of lecture 5
03:27 Equivariant homotopy sets of orthogonal spaces
09:22 Invariant description of elements in equivariant homotopy sets
13:24 Restriction maps on equivariant homotopy sets
17:29 Inner automorphisms act as the identity
25:51 Fixed points of orbits by free actions
32:18 Equivariant homotopy sets of global classifying spaces
40:12 Example: abelian compact Lie groups
42:49 Example: the orthogonal group O(n)
44:03 Stabilization of equivariant homotopy classes
46:04 Unstable versus stable tautological class
47:12 Stabilization commutes with restriction
48:20 Transfers of stabilizations
52:27 The 0-th equivariant homotopy groups of global suspension spectra
56:14 The 0-th equivariant homotopy groups of G-equivariant suspension spectra
59:08 The reduction argument
1:02:54 The special case of global classifying spaces

For more information about the class, such as the topics, prerequisites, or references, visit the webpage:

Special thanks to JDLP for the post production of this video