Projective elements in \(K\)-theory and self maps of \(\Sigma CP^{\infty}\)
type: Item
instance of: scholarly article
title: Projective elements in \(K\)-theory and self maps of \(\Sigma CP^{\infty}\)
zbMATH Open document ID: 0924.55002
author: Kaoru Morisugi
published in: Journal of Mathematics of Kyoto University
publication date: 1999-11-11T00:00:00Z
review text: The author works in the homotopy category of based spaces and based maps. Given a space \(X\), let the reduced \(K\)-theory be denoted by \(K(X)\) and the homology group of integral coefficients by \(H_*(X)\). Let \(\mathbb{C}\text{P}^\infty\) be the infinite-dimensional complex projective space. Let \(\eta\) be the canonical line bundle over \(\mathbb{C}\text{P}^\infty\) and \(i:\mathbb{C}\text{P}^\infty\to \text{BU}\) be the classifying map of the virtual bundle \(\eta-1\). Since BU has a loop space structure, there exists a unique extension of \(i\) to the loop map \(j:\Omega\Sigma \mathbb{C}\text{P}^\infty\to \text{BU}\). In this paper, the author investigates the following problems: Given an element \(\alpha\in K(X)\), when does there exist a lift \(\widehat\alpha\in [X,\Omega\Sigma \mathbb{C}\text{P}^\infty]\) such that \(j_*(\widehat\alpha)= \alpha\)? If \(\alpha\) has a lift, how can we construct the lift \(\widehat\alpha\)? Define \(\text{PK}(X)= \{\alpha\in K(X)\mid\exists\widehat\alpha\in [X,\Omega\Sigma \mathbb{C}\text{P}^\infty]\) such that \(j_*(\widehat\alpha)= \alpha\}\). If an element \(\alpha\in K(X)\) belongs to \(\text{PK}(X)\), one says that \(\alpha\) is projective. The significance of the above problem is as follows: The James splitting theorem [\textit{I. M. James}, ``The topology of Stiefel manifolds'', Lect. Note Series 24 (1976; Zbl 0337.55017)] implies that there exists a loop map \(\theta: \text{BU}\to \Omega^\infty \Sigma^\infty\mathbb{C}\text{P}^\infty\) such that \(\theta\circ j= E^\infty: \Omega\Sigma\mathbb{C}\text{P}^\infty\to \Omega^\infty\Sigma^\infty \mathbb{C}\text{P}^\infty\). Therefore, given an element \(\alpha\in K(X)\), the stable map, \(\text{ad }j\circ(\theta(\alpha)): \Sigma^\infty X\to\Sigma^\infty \mathbb{C}\text{P}^\infty\) can be considered. Using the information of \(K(X)\), the induced homomorphism [\textit{C. A. McGibbon}, Trans. Am. Math. Soc. 271, 325-346 (1982; Zbl 0491.55014); \textit{K. Morisugi}, Publ. Res.…
reviewed by: Ioan Pop
Mathematics Subject Classification ID: 55N15
zbMATH DE Number: 1295046
zbMATH Keywords: projective element
zbMATH Keywords: lift
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250518164
OpenAlex ID: W1923207764
DOI: 10.1215/KJM/1250518164
description: scientific article; zbMATH DE number 1295046