Algebraic cycles and infinite loop spaces
type: Item
instance of: scholarly article
title: Algebraic cycles and infinite loop spaces
zbMATH Open document ID: 0797.55006
DOI: 10.1007/BF01244311
author: Charles P. Boyer
author: H. Blaine jun. Lawson
author: Paulo Lima-Filho
author: Benjamin M. Mann
author: Marie-Louise Michelsohn
published in: Inventiones Mathematicae
publication date: 1994-10-19T00:00:00Z
full work available at URL: https://eudml.org/doc/144131
review text: Let \(\mathbb{Z} \times BU\) be the space representing complex \(K\)-theory, and let \(K(\mathbb{Z},\text{even}) = \prod^ \infty_{j = 0} K(\mathbb{Z},2j)\) be the product of Eilenberg-MacLane spaces representing the functor \(\prod^ \infty_{j = 0} H^{2j}(-)\). The authors construct an infinite loop space structure on \(K(\mathbb{Z},\text{even})\) such that the total Chern class \(c: \mathbb{Z} \times BU \to K(\mathbb{Z},\text{even})\) is an infinite loop map, and they prove a similar result for the total Stiefel-Whitney class. The constructions use algebraic cycles [see \textit{H. B. Lawson jun.}, Ann. Math., II. Ser. 129, No. 2, 253-291 (1989; Zbl 0688.14006)]. The \(H\)-space structure induced on \(K(\mathbb{Z},\text{even})\) by the infinite loop space structure represents the cup product. There is another infinite loop space structure on \(K(\mathbb{Z},\text{even})\) representing the cup product due to \textit{G. Segal} [Quart. J. Math., Oxford II. Ser. 26, 289-293 (1975; Zbl 0321.55017)]. But the infinite loop space structure used here is different from Segal's; indeed, with Segal's structure, \(c\) is not an infinite loop map.
Mathematics Subject Classification ID: 55P47
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 14C25
Mathematics Subject Classification ID: 14C05
zbMATH DE Number: 549659
zbMATH Keywords: complex \(K\)-theory
zbMATH Keywords: infinite loop space
zbMATH Keywords: total Chern class
zbMATH Keywords: infinite loop map
zbMATH Keywords: total Stiefel-Whitney class
zbMATH Keywords: algebraic cycles
reviewed by: Richard John Steiner
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 549659