An almost classification of compact Lie groups with Borsuk-Ulam properties
type: Item
instance of: scholarly article
title: An almost classification of compact Lie groups with Borsuk-Ulam properties
zbMATH Open document ID: 0685.57021
author: Wacław Marzantowicz
published in: Pacific Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: We say that a compact Lie group G has the Borsuk-Ulam property in the weak sense if for every orthogonal representation V of G and every G- equivariant map f: S(V)\(\to S(V)\), \(V^ 0=\{0\}\), of the unit sphere we have deg \(f\neq 0\). We say that G has the Borsuk-Ulam property in the strong sense if for any two orthogonal representations V, W of G with dim W\(=\dim V\) and \(W^ 0=V^ 0=\{0\}\) and every G-equivariant map f: S(V)\(\to S(W)\) of the unit sphere, we have deg \(f\neq 0\). In this paper a complete classification, up to isomorphism, of groups with the weak Borsuk-Ulam property is given. A classification of groups with the strong Borsuk-Ulam property does not cover nonabelian p-groups with all elements of order p. In fact, we deal with a more general definition admitting a nonempty fixed point set of G on the sphere S(V).
Mathematics Subject Classification ID: 57S15
Mathematics Subject Classification ID: 55M20
zbMATH DE Number: 4122862
zbMATH Keywords: compact Lie group
zbMATH Keywords: Borsuk-Ulam property
zbMATH Keywords: equivariant map
zbMATH Keywords: weak Borsuk- Ulam property
zbMATH Keywords: strong Borsuk-Ulam property
zbMATH Keywords: degree of maps
MaRDI profile type: Publication
Wikidata QID: Q115230754
full work available at URL: pjm.1990.144.299
OpenAlex ID: W1984964382
DOI: 10.2140/PJM.1990.144.299
description: scientific article; zbMATH DE number 4122862