On \(Sp(2)\) and \(Sp(2)\cdot Sp(1)\)-structures in 8-dimensional vector bundles
type: Item
instance of: scholarly article
title: On \(Sp(2)\) and \(Sp(2)\cdot Sp(1)\)-structures in 8-dimensional vector bundles
zbMATH Open document ID: 0896.57015
DOI: 10.5565/PUBLMAT_41297_05
author: Martin Čadek
author: Jiří Vanžura
published in: Publicacions Matemàtiques
publication date: 1998-09-22T00:00:00Z
review text: Given an oriented \(8\)-plane bundle \(\xi\) over a CW-complex, the authors associate to it a certain other vector bundle \(\zeta\). They come to the conclusion that the structure group \(SO(8)\) of \(\xi\) can be reduced (a) to \(Sp(2)\cdot Sp(1)\) if and only if \(\zeta\) contains an oriented \(3\)-plane bundle, and (b) to \(Sp(2)\) if and only if \(\zeta\) has \(3\) everywhere linearly independent sections. In their main result, for \(\xi\) an oriented \(8\)-plane bundle over a closed connected spin \(8\)-manifold, they present the condition of (b) expressed in terms of Stiefel-Whitney classes, Pontryagin classes, and the Euler class of \(\xi\).
reviewed by: Július Korbaš
Mathematics Subject Classification ID: 57R22
Mathematics Subject Classification ID: 57R15
Mathematics Subject Classification ID: 57R25
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 1146190
zbMATH Keywords: \(Sp(2)\)-structure
zbMATH Keywords: \(Sp(2)\cdot Sp(1)\)-structure
zbMATH Keywords: \(8\)-dimensional vector bundle
zbMATH Keywords: hyper-Kähler manifold
zbMATH Keywords: quaternion-Kähler manifold
zbMATH Keywords: linearly independent sections
zbMATH Keywords: Cayley numbers
zbMATH Keywords: principle of triality
zbMATH Keywords: Stiefel-Whitney class
zbMATH Keywords: Pontryagin class
zbMATH Keywords: Chern class
zbMATH Keywords: Euler class
zbMATH Keywords: 8-plane bundle
MaRDI profile type: Publication
full work available at URL: publmat 41297 05
OpenAlex ID: W1998888358
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description: scientific article; zbMATH DE number 1146190