A remark on the Segal-Becker theorem
type: Item
instance of: scholarly article
title: A remark on the Segal-Becker theorem
zbMATH Open document ID: 0574.55015
author: Kazumoto Kozima
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1984-01-01T00:00:00Z
review text: Beginning with the natural inclusion \(j: {\mathbb{C}}P^{\infty}\to BU\), one may construct - using Bott periodicity - \(\xi\) : Q(BU)\(\to BU\), and then \(\lambda\) : Q(\({\mathbb{C}}P^{\infty})\to BU\) as the map \(\xi\) \(\circ Q(j)\). It is known, from results of \textit{J. C. Becker} [Lect. Notes Math. 428, 132-143 (1974; Zbl 0303.55002)] and \textit{G. Segal} [Q. J. Math., Oxf. II. Ser. 24, 1-5 (1973; Zbl 0266.55009)] that \(\lambda\) has a right-inverse written s. The purpose of the present paper is to find such a map s: BU\(\to Q({\mathbb{C}}P^{\infty})\) so that \(s\circ j\) is homotopic to the inclusion \({\mathbb{C}}P^{\infty}\to Q({\mathbb{C}}P^{\infty})\). The methods use the work of \textit{G. Brumfield} and \textit{I. Madsen} [Invent. Math. 32, 133-169 (1976; Zbl 0319.55027)].
Mathematics Subject Classification ID: 55R50
Mathematics Subject Classification ID: 55R12
Mathematics Subject Classification ID: 55P42
Mathematics Subject Classification ID: 55P47
Mathematics Subject Classification ID: 55N15
zbMATH DE Number: 3917261
zbMATH Keywords: stable homotopy theory of infinite complex projective space
zbMATH Keywords: unitary K- theory
zbMATH Keywords: transfer
reviewed by: Donald W. Kahn
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195181412
OpenAlex ID: W2053046396
cites work: Infinite Loop Spaces (AM-90)
cites work: Q4057799
cites work: The transfer map and fiber bundles
cites work: Evaluation of the transfer and the universal surgery classes
cites work: Transfer in generalized cohomology theories
cites work: THE STABLE HOMOTOPY OF COMPLEX PROJECTIVE SPACE
DOI: 10.2977/PRIMS/1195181412
description: scientific article; zbMATH DE number 3917261