On complex structures in 8-dimensional vector bundles
type: Item
instance of: scholarly article
title: On complex structures in 8-dimensional vector bundles
zbMATH Open document ID: 0902.57029
author: Martin Čadek
author: Jiří Vanžura
published in: Manuscripta Mathematica
publication date: 1998-06-24T00:00:00Z
full work available at URL: https://eudml.org/doc/156379
review text: Let \(\xi\) be an oriented real \(8\)-plane bundle over a closed connected oriented smooth \(8\)-dimensional manifold \(M\). Supposing that the Stiefel-Whitney classes \(w_2(\xi)\) and \(w_2(M)\) vanish, the authors reduce the question of whether \(\xi\) has a complex structure to the problem of whether a certain associated \(8\)-plane bundle contains an oriented \(2\)-plane bundle. Then they use results from their earlier paper [Colloq. Math. 70, No. 1, 25-40 (1996; Zbl 0844.57026)] to derive explicit necessary and sufficient conditions for the existence of complex structures on \(\xi\) in terms of the cohomology ring of \(M\) and characteristic classes of \(\xi\) and \(M\). Without requiring that \(w_2(M) = 0\), \textit{T. Heaps} [Topology 9, 111-119 (1970; Zbl 0187.20405)] found similar necessary and sufficient conditions for the existence of complex structures on the tangent bundle of \(M\).
reviewed by: Július Korbaš
Mathematics Subject Classification ID: 57R22
Mathematics Subject Classification ID: 57R25
Mathematics Subject Classification ID: 55R25
Mathematics Subject Classification ID: 22E99
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 1143110
zbMATH Keywords: \(8\)-dimensional vector bundle
zbMATH Keywords: \(8\)-dimensional manifold
zbMATH Keywords: spin structure
zbMATH Keywords: complex structure
zbMATH Keywords: characteristic class
zbMATH Keywords: 8-plane bundle
MaRDI profile type: Publication
OpenAlex ID: W4247893054
DOI: 10.1007/S002290050032
description: scientific article; zbMATH DE number 1143110