Stable extendibility of vector bundles over real projective spaces and bounds for the Schwarzenberger numbers \(\beta(k)\)
type: Item
instance of: scholarly article
title: Stable extendibility of vector bundles over real projective spaces and bounds for the Schwarzenberger numbers \(\beta(k)\)
zbMATH Open document ID: 1062.55016
author: Teiichi Kobayashi
author: Toshio Yoshida
published in: Hiroshima Mathematical Journal
publication date: 2004-12-29T00:00:00Z
review text: Let \(X\) be a space, \(A\) a subspace and \(i: A\rightarrow X\) the inclusion map. Let \(F\) denote the field of real or complex numbers. A \(k\)-dimensional \(F\)-vector bundle \(\zeta\) over \(A\) is called stably extendible to \(X\) if there exists a \(k\)-dimensional \(F\)-vector bundle \(\eta\) over \(X\) such that \(i^{*}\eta\) is stably equivalent to \(\zeta .\) The authors consider the problem of stable extendibility of real (resp. complex) vector bundles over \(\mathbb R P^n\) to real (resp. complex) vector bundles over \(\mathbb R P^m\) (\(m>n\)). In each case they give four equivalent conditions characterizing when a stable real (resp. complex) \(k\)-dimensional vector bundle \(\zeta\) over \(\mathbb R P^n\) is stably extendible to \(\mathbb R P^m\) for every \(m\geq n.\) The conditions are formulated by means of the Schwarzenberger number \({\beta}(k)\) and the function \({\phi}(n)\). These arithmetic functions are defined as follows: Let \(i=(2a+1)2^{{\nu}(i)}\) then \({\beta}(k)=\min\{i-{\nu}(i)-1 \mid k<i\},\) \({\phi}(n)\) is the number of integers \(s\) with \(0 < s \leq n\) and \(s\) is congruent to \(0,1,2,4\mod 8.\)
reviewed by: Piotr Krasoń
Mathematics Subject Classification ID: 55R50
Mathematics Subject Classification ID: 55N15
zbMATH DE Number: 2123907
zbMATH Keywords: vector bundle
zbMATH Keywords: stable extendibility
zbMATH Keywords: real projective space
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1150997981
OpenAlex ID: W1919380109
DOI: 10.32917/HMJ/1150997981
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description: scientific article; zbMATH DE number 2123907