Symmetric products, subgroup lattices and filtrations of global K-theory

Abstract

This thesis consists of two projects in equivariant stable homotopy theory. In the first we study the rational homotopy groups of symmetric products of the G-sphere spectrum and show that they are naturally isomorphic to the rational homology groups of certain subcomplexes of the subgroup lattice of G. In the second we investigate global equivariant versions of spectrum level filtrations introduced by Arone and Lesh which interpolate between topological/algebraic K-theory and the Eilenberg-MacLane spectrum for the integers. We determine the global homotopy type of the subquotients and use this description to obtain algebraic formulas for filtrations of representation rings that arise on 0-th homotopy groups.