Units of ring spectra and their traces in algebraic K-theory

Abstract

Let GL_1(R) be the units of a commutative ring spectrum R. In this paper we identify the composition BGL_1(R)->K(R)->THH(R)->Ω^{\infty}(R), where K(R) is the algebraic K-theory and THH(R) the topological Hochschild homology of R. As a corollary we show that classes in π_{i-1}(R) not annihilated by the stable Hopf map give rise to non-trivial classes in K_i(R) for i\geq 3.

Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper16.abs.html

Cited by in corpus (26)

• Diagram spaces and symmetric spectra
• Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory
• Topological Hochschild homology of Thom spectra and the free loop space
• Diagram spaces, diagram spectra, and spectra of units
• Higher topological Hochschild homology of Thom spectra
• The character map in (twisted differential) non-abelian cohomology
• On the algebraic K-theory of the complex K-theory spectrum
• Group completion and units in I-spaces
• Logarithmic structures on topological K-theory spectra
• Rational algebraic K-theory of topological K-theory
• Infinite loop spaces and nilpotent K-theory
• Spectra of units for periodic ring spectra and group completion of graded E-infinity spaces
• Unit spectra of K-theory from strongly self-absorbing C*-algebras
• $K$-theory of Hermitian Mackey functors and a reformulation of the Novikov Conjecture
• Multiplicative parametrized homotopy theory via symmetric spectra in retractive spaces
• The factorization theory of Thom spectra and twisted non-abelian Poincaré duality
• Spectra associated to symmetric monoidal bicategories
• Generalized Thom spectra and their topological Hochschild homology
• Virtual vector bundles and graded Thom spectra
• Periodic points and topological restriction homology
• Exponential Functors, R-Matrices and Twists
• Bundles of spectra and algebraic K-theory
• Interpreting the Bokstedt smash product as the norm
• Equivariant higher Dixmier-Douady Theory for circle actions on UHF-algebras
• Bundles of strongly self-absorbing $C^{\ast }$-algebras with a Clifford grading
• The weak b-principle: Mumford conjecture