Spheres and symmetry: Borsuk's antipodal theorem
type: Item
instance of: scholarly article
title: Spheres and symmetry: Borsuk's antipodal theorem
zbMATH Open document ID: 0795.55004
DOI: 10.12775/TMNA.1993.004
published in: Topological Methods in Nonlinear Analysis
publication date: 1994-08-29T00:00:00Z
review text: This nicely-written survey article concerns the fundamental discovery of Borsuk that a continuous function from the \(n\)-sphere to Euclidean space of the same dimension must send at least one pair of antipodal points of the sphere to the same point in the Euclidean space. The author not only lists eight equivalent, and impressively diverse, equivalent forms of Borsuk's result, he also sketches three strikingly different approaches to proving it. A brief, interesting exploration of some of the limits to extending Borsuk's theorem is followed by the discussion of a particularly fruitful direction of generalization: to group actions on spheres. After an exceedingly condensed section on index theories, the paper closes with eight well-chosen and clearly-described applications of Borsuk's discovery. This relatively brief survey includes a useful 80- item list of references, but a reader who wishes a more detailed and comprehensive survey of the same topic should consult the author's article in [Méthodes topologiques en analyse non linéaire, Sémin. Math. Supér., Sémin. Sci. OTAN (NATO Adv. Study Inst.) 95, 166-235 (1985; Zbl 0573.55003)].
Mathematics Subject Classification ID: 55M30
zbMATH DE Number: 482104
zbMATH Keywords: Lusternik-Schnirelmann category
zbMATH Keywords: group action
zbMATH Keywords: Borsuk-Ulam theorem
author: Heinrich Steinlein
reviewed by: Robert F. Brown
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 482104