Two topics in stable homotopy theory

Abstract

We give a definition of a norm functor from H-Mackey functors to G-Mackey functors for G a finite group and H a subgroup of G. We check that this agrees with the construction of Mazur in the case G cyclic of prime power order and also with the topological definition of norm, which has an algebraic presentation due to Ullman. We then use this norm functor to give a characterization of Tambara functors as monoids of an appropriate flavor.

The second chapter is part of a joint project with Andrew Baker. We consider what happens when we take the sphere spectrum, and kill elements of homotopy in an E_∞ fashion. This process starts with the element 2 and is repeated in order to kill all higher homotopy groups. We provide methods for identifying spherical classes and for understanding the Dyer-Lashof action at each step of the construction. We outline how this construction might be used to compute the André-Quillen homology of Eilenberg-MacLane spectra considered as algebras over the sphere spectrum.