In spring 2021, I ran a hybrid seminar/course on the proof of the non-existence of elements of Kervaire invariant one, based on Hill-Hopkins-Ravenel. We met on Mondays 1 - 3 p.m.
In this course, we develop the necessary tools and then go through the Hill-Hopkins-Ravenel proof of the Kervaire invariant one problem. In the first half, we develop the equivariant stable homotopy theory. In the second half, we apply equivariant stable homotopy theory to give a complete proof of the Kervaire invariant one problem.
Schedule:
First part:
Introduction. Notes.
Enriched category, model category, and abstract homotopy theory we need. Jack Davies. Notes. Exercises.
G-spectra: model structures, Wirthmuller isomorphism. Yuqing Shi. Notes. Exercises.
G-spectra: All fixed points, the norm functor. Hana Jia Kong. Notes.
Mackey functors, homotopy of G-spectra and equivariant Eilenberg-MacLane spectra. Leonard Tokic. Notes.
MUR as a cofibrant commutative C2-ring spectrum. Jaco Ruit. Notes.
Second part:
The slice filtration and the slice tower. Notes.
Underlying homotopy of BPG, twisted monoid rings, and equivariant quotient modules. Notes.
The slice tower of BPG and its spectral sequence. Notes.
Slice differentials. Notes.
The periodicity theorem and \Omega. Notes.
The gap theorem, the homotopy fixed points theorem, and some equivariant commutative ring spectra. Notes.
The detection theorem. Notes.
Guest lecture by Mike Hill: Future directions and open problems. Notes.