The local defect index up to finite ambiguity
type: Item
instance of: scholarly article
title: The local defect index up to finite ambiguity
zbMATH Open document ID: 0996.55014
DOI: 10.1016/S0166-8641(01)00064-5
author: S. Bechtluft-Sachs
author: Marco Hien
published in: Topology and its Applications
publication date: 2002-05-15T00:00:00Z
review text: Let \(E\) be a \(q\)-dimensional oriented vector bundle with \(q\geq 3\) over a \((p+1)\)-dimensional cell complex \(X\) and let \(\sigma\) be a nowhere vanishing cross section over the \(p\)-skeleton \(X_p\) of \(X\). Then we have a well-known cohomology class \(c(E,\sigma)\) in \(H^{p+1} (X;\pi_p(S^{q-1}))\) which is the primary obstruction to extending \(\sigma|X_{p-1}\) to \(X\). The authors consider especially the rational class of this obstruction. However we know that the coefficient group is infinite only if \(p=q-1\) or \(p=4\ell-1\), \(q=2\ell+1\) and further in the first case \(c(E,\sigma)\) is just the Euler class of \(E\). So the present paper deals with the second case. The result is \(p_\ell(E)-e( \sigma)^2= (-1)^\ell 4h_*(c(E,\sigma)) \in H^{4\ell}(X:\mathbb{Q})\) where \(p_\ell (E)\) denotes the \(\ell\)th Pontryagin class of \(E\), \(e(\sigma)\) the Euler class of the complementary bundle of \(\sigma\) and \(h:\pi_{4\ell-1}(S^{2\ell}) \to\mathbb{Z} \subset\mathbb{Q}\) the Hopf invariant.
reviewed by: Haruo Minami
Mathematics Subject Classification ID: 55R25
Mathematics Subject Classification ID: 55S40
zbMATH DE Number: 1741952
zbMATH Keywords: Hopf invariant
MaRDI profile type: Publication
cites work: Topological properties of ordinary nematics in 3-space
cites work: Q4192669
description: scientific article; zbMATH DE number 1741952