There are no phantom cohomology operations in K-theory
type: Item
instance of: scholarly article
title: There are no phantom cohomology operations in K-theory
zbMATH Open document ID: 0551.55012
author: D. W. Anderson
published in: Pacific Journal of Mathematics
publication date: 1983-01-01T00:00:00Z
review text: The main theorem of this paper deals with cohomology operations in connected K-theories. A stable cohomology operation between cohomology theories \(h^*_ 1\to h_ 2^*\) is called phantom if it induces the zero operation when restricted to finite complexes. An \(\Omega\)- represented operation is called completely phantom if it vanishes on all CW complexes. It is proved that there are no (non-zero) phantom or completely phantom operations between the various connected K-theories. In the course of proving this theorem the author performs a great service to homotopy theorists by going through his clear and precise versions of (inverse) limit theories for moduli spaces and spectra, the Atiyah- Hirzebruch and Dold spectral sequences and stable homotopy theory. The latter is studied in the framework of Quillen's axiomatic homotopy theory. As the author has his particular applications in mind, clarity and accuracy replace complete generality.
Mathematics Subject Classification ID: 55S25
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 55N20
Mathematics Subject Classification ID: 55P42
Mathematics Subject Classification ID: 18A30
Mathematics Subject Classification ID: 55U35
zbMATH DE Number: 3877975
zbMATH Keywords: Atiyah-Hirzebruch spectral sequence
zbMATH Keywords: cohomology operations in connected K-theories
zbMATH Keywords: phantom
zbMATH Keywords: completely phantom operations
zbMATH Keywords: Dold spectral sequences
zbMATH Keywords: axiomatic homotopy theory
MaRDI profile type: Publication
full work available at URL: pjm.1983.107.279
OpenAlex ID: W2074692608
DOI: 10.2140/PJM.1983.107.279
description: scientific article; zbMATH DE number 3877975