Higher coherences for equivariant K-theory

Abstract

A bstract . Let G be a compact Lie group. We show that concepts of operator theory can be used to define an E ∞ -ring spectrum representing G -equivariant K -theory. In addition we construct an E ∞ -model for the G -equivariant Atiyah-Bott-Shapiro orientation M Spin c → K . INTRODUCTION About twenty-five years ago May, Quinn and Ray introduced the concept of E ∞ -ring spectra [16, Chapter IV]. The definition was motiviated by the fact that there is no way to construct an internal smash product on the category of ordinary spectra and functions between them in such a way that the smash product would equip this category of spectra with a symmetric monoidal product. Of course there is the well-defined smash product on the homotopy category of spectra. An E ∞ -structure on a commutative “homotopy ring spectrum” R or on a module M over it essentially guarantees that the homotopy multiplications R ∧ R → R and R ∧ M → M satisfy “all relevant algebraic relations”. For example, E ∞ -structures allow to define the smash product of two E ∞ -module spectra over an E ∞ -ring spectrum which then again is an E ∞ -module spectrum over the E ∞ -ring spectrum which then again will be an E ∞ -module spectrum over the E ∞ -ring spectrum. Recently however several people suceeded in defining a symmetric monoidal smash product on certain categories of spectra. Of course, for doing so one needs to put some extra structure on the spectra.