Book contents
- Frontmatter
- Contents
- Preface
- The development of structured ring spectra
- Compromises forced by Lewis's theorem
- Permutative categories as a model of connective stable homotopy
- Morita theory in abelian, derived and stable model categories
- Higher coherences for equivariant K-theory
- (Co-)Homology theories for commutative (S-)algebras
- Classical obstructions and S-algebras
- Moduli spaces of commutative ring spectra
- Cohomology theories for highly structured ring spectra
- Index
Higher coherences for equivariant K-theory
Published online by Cambridge University Press: 23 October 2009
- Frontmatter
- Contents
- Preface
- The development of structured ring spectra
- Compromises forced by Lewis's theorem
- Permutative categories as a model of connective stable homotopy
- Morita theory in abelian, derived and stable model categories
- Higher coherences for equivariant K-theory
- (Co-)Homology theories for commutative (S-)algebras
- Classical obstructions and S-algebras
- Moduli spaces of commutative ring spectra
- Cohomology theories for highly structured ring spectra
- Index
Summary
Abstract. 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 MSpinc → 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.
- Type
- Chapter
- Information
- Structured Ring Spectra , pp. 87 - 114Publisher: Cambridge University PressPrint publication year: 2004
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