Differentiable fibrations of the (2n-1)-sphere by great (n-1)-spheres and their coordinatization over quasifields
type: Item
instance of: scholarly article
title: Differentiable fibrations of the (2n-1)-sphere by great (n-1)-spheres and their coordinatization over quasifields
zbMATH Open document ID: 0648.53042
DOI: 10.1007/BF03322382
author: Hermann Hähl
published in: Results in Mathematics
publication date: 1987-01-01T00:00:00Z
review text: The author presents a variation of the proof of the following theorem of \textit{H. Gluck}, \textit{F. Warner} and \textit{C. T. Yang}: Theorem. A locally trivial differentiable fibre bundle with total space \(S^{2n-1}\) whose fibres are great (n-1)-spheres is topologically equivalent to one of the classical Hopf-fibrations [cf. Duke Math. J. 50, 1041-1076 (1983; Zbl 0534.53039)]. With each fibering \(\pi\) of \(S^{2n-1}\) by great (n-1)- spheres is associated a quasifield with additive group (\({\mathbb{R}}^ n,+)\) and linear left-multiplication. Relations to the theory of translation planes are briefly indicated. A characterization of locally trivial differentiable fibre bundles by differentiability properties of the associated quasifield is given. Finally Buchanan's theorem can be applied to the characteristic map of the associated differentiable quasifield. Cerf's result on diffeomorphisms of S 3 is used to show that even differentiable equivalence holds in the theorem in the case \(n=4\).
Mathematics Subject Classification ID: 53C99
Mathematics Subject Classification ID: 51A40
Mathematics Subject Classification ID: 55R25
zbMATH DE Number: 4058361
zbMATH Keywords: differentiable fibre bundle
zbMATH Keywords: Hopf-fibrations
zbMATH Keywords: differentiable quasifield
MaRDI profile type: Publication
cites work: On the nonexistence of elements of Hopf invariant one
cites work: Quasigroups. I
cites work: 4-dimensionale Translationsebenen mit kommutativer Standgruppe
cites work: Zur Topologie der Projektiven Ebenen über reellen Divisionsalgebren
cites work: Sur les difféomorphismes de la sphère de dimension trois \(\Gamma_ 4 = 0\)
cites work: Great circle fibrations of the three-sphere
cites work: Division algebras, fibrations of spheres by great spheres and the topological determination of space by the gross behavior of its geodesics
cites work: Fibrations of spheres by great spheres over division algebras and their differentiability
cites work: Eine Klasse von achtdimensionalen lokalkompakten Translationsebenen mit großen Scherungsgruppen
cites work: Q1084338
cites work: Sechzehndimensionale lokalkompakte Translationsebenen mit Spin(7) als Kollineationsgruppe. (Sixteen-dimensional locally compact translation planes with Spin(7) as their collineation group)
cites work: On topological Blaschke conjecture. II: Nonexistence of Blaschke structure on fake quaternion projective planes
cites work: Q3736501
cites work: Division algebras and fibrations of spheres by great spheres
full work available at URL: https://doi.org/10.1007/bf03322382
OpenAlex ID: W1967666442
description: scientific article; zbMATH DE number 4058361