A Borsuk-Ulam theorem for orthogonal \(T^ k\) and \(Z^ r_ p\) actions and applications
type: Item
instance of: scholarly article
title: A Borsuk-Ulam theorem for orthogonal \(T^ k\) and \(Z^ r_ p\) actions and applications
zbMATH Open document ID: 0706.55001
DOI: 10.1016/0022-247X(89)90275-8
published in: Journal of Mathematical Analysis and Applications
publication date: 1989-01-01T00:00:00Z
review text: The author first gives, for a compact Lie group G, the notion of Borsuk- Ulam property in the weak (resp. strong) sense A (in short, property IA (resp. IIA)) and then proves that the k-dimensional torus group \(T^ k\) and the p-torus \(Z^ r_ p\) have property IIA, which naturally implies IA. The classical Borsuk-Ulam theorem asserts that \(G=Z_ 2\) has property IA. The author introduces the notion of geometric G-index \(\gamma_ G(X)\) for G-spaces X. This index is used to estimate the number of multiple critical points for functions invariant under a \(T^ k\) or \(Z^ r_ p\) action on a compact smooth manifold without boundary.
Mathematics Subject Classification ID: 55M20
Mathematics Subject Classification ID: 55M35
Mathematics Subject Classification ID: 57S10
Mathematics Subject Classification ID: 57S17
zbMATH DE Number: 4159619
zbMATH Keywords: torus actions
zbMATH Keywords: actions of products of \({\mathbb{Z}}/p\)
zbMATH Keywords: Borsuk-Ulam property
zbMATH Keywords: k-dimensional torus
zbMATH Keywords: p-torus
author: Wacław Marzantowicz
reviewed by: Akihiko Morimoto
MaRDI profile type: Publication
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full work available at URL: 0022-247x(89)90275-8
OpenAlex ID: W143088820
description: scientific article; zbMATH DE number 4159619