This will be an advanced topics course on equivariant homotopy theory. Over the term we will develop modern foundations required to do homotopy theory equivariantly, i.e., with group actions. Although there are no strict prerequisites for the course, some topics that are usually taught in a course in homotopy theory will be discussed briefly or treated as a black box.
To develop these foundations I will begin by discussing two foundational frameworks: model categories and infinity-categories. It turns out there are many different possible equivariant homotopy theories and we will construct them generally as model categories and then take their associated infinity-categories. From here we can construct Eilenberg-MacLane spaces and Postnikov towers.
We will then consider the equivariant analogues of ordinary (co)homology and bundle theory. We will then consider the equivariant Pontryagin-Thom construction from the classical viewpoint. This leads to the equivariant analogue of Spanier-Whitehead duality and motivates the construction of genuine equivariant spectra.
We will construct G-spectra from a calculus of functors approach and compare to orthogonal spectra. After this the hope is to discuss parametrized spectra (locally constant sheaves of spectra) and spend some time on equivariant duality theory.
Further additional topics may include: equivariant K-theory and Bott periodicity, global equivariant homotopy theory, the norm construction and applications, equivariant rational unstable homotopy theory, and the Segal conjecture.
- Course begins: November 7th, 2013
- Course time: 12:00-14:00 Thursday + Friday
- Course location: M 103
- Exercise session: 18:00-20:00 Thursday
- Exercise session location: M 206 (if there are too many people we will move to M 103)
- Office hours by appointment in M 206
- There will be an oral exam at the end of the term (scheduled by appointment.)