On splitting topological \(R^ 4\)-bundles
type: Item
instance of: scholarly article
title: On splitting topological \(R^ 4\)-bundles
zbMATH Open document ID: 0756.57009
DOI: 10.1016/0166-8641(92)90152-P
published in: Topology and its Applications
publication date: 1992-09-27T00:00:00Z
review text: Suppose \(\xi\) is an \(\mathbb{R}^ 4\)-bundle over a Poincaré complex of formal dimension at most 4, and \(\alpha\) is either a line or a plane bundle over \(X\). This paper considers conditions involving characteristic classes of \(\xi\) and \(\alpha\) which determine when there is a Whitney sum decomposition \(\xi\approx \alpha\oplus\beta\), for some bundle \(\beta\). As an example of an application, it is shown that the tangent bundle of a closed connected 4-manifold splits off a given line bundle if and only if both the Euler number and the Kirby-Siebenmann number of the manifold vanish.
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 57R22
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 57R19
Mathematics Subject Classification ID: 55R25
zbMATH DE Number: 60705
zbMATH Keywords: \(\mathbb{R}^ 4\)-bundle over a Poincaré complex of formal dimension at most 4
zbMATH Keywords: characteristic classes
zbMATH Keywords: tangent bundle of a closed connected 4- manifold
zbMATH Keywords: Euler number
zbMATH Keywords: Kirby-Siebenmann number
author: Carl H. Brans
author: Duane Randall
reviewed by: David B. Gauld
MaRDI profile type: Publication
cites work: Classification of oriented sphere bundles over a 4-complex
cites work: On the classification of topological 4-manifolds with finite fundamental group
cites work: Felder von Flächenelementen in 4-dimensionalen Mannigfaltigkeiten. (Plane element fields in 4-dimensional manifolds)
cites work: Q4105006
cites work: Foundational Essays on Topological Manifolds, Smoothings, and Triangulations. (AM-88)
cites work: Concordance and bordism of line fields
cites work: Two-plane sub-bundles of nonorientable real vector-bundles
cites work: CAT 2-FIELDS ON NONORIENTABLE CAT MANIFOLDS
cites work: Q3812940
cites work: Fields of tangent 2-planes on even-dimensional manifolds
cites work: Q4192669
description: scientific article; zbMATH DE number 60705