Bourgin-Yang version of the Borsuk-Ulam theorem for \(\mathbb Z_{p^{k}}\)-equivariant maps
type: Item
instance of: scholarly article
title: Bourgin-Yang version of the Borsuk-Ulam theorem for \(\mathbb Z_{p^{k}}\)-equivariant maps
zbMATH Open document ID: 1262.55001
author: Wacław Marzantowicz
author: Denise De Mattos
author: Edivaldo L. dos Santos
published in: Algebraic \& Geometric Topology
publication date: 2013-01-07T00:00:00Z
review text: Let \(V\), \(W\) be two orthogonal representations of a cyclic group \(G\) of prime power order \(p^k\) such that \(V^G=W^G=\{0\}\), let \(S(V)\) be the unit sphere of \(V\) and let \(F:S(V)\to W\) be a \(G\)-equivariant map. Let \(Z_f=f^{-1}(\{0\})\). The authors give a lower bound for the (covering) dimension \(\dim Z_f=\dim(Z_f/G)\). Denote by \(\mathcal{A}_{S(V)}\) the set of all \(G\)-orbits of \(S(V)\) and call \(\mathcal{A}_{m,n}:=\{G/H|\, H\subset G: m\leq|H|\leq n\}\) (where \(1\leq m\leq n\leq p^{k-1}\) are two powers of \(p\)) and assume that both \(\mathcal{A}_{S(V)}\) and \(\mathcal{A}_{S(W)}\) are subsets of \(\mathcal{A}_{m,n}\). In this situation, the authors prove that \(\dim Z_f\geq2\bigl(\bigl[\frac{(\dim(V)-1)m}{n}\bigr]-\dim W\bigr)\).
Mathematics Subject Classification ID: 55M20
Mathematics Subject Classification ID: 55M35
Mathematics Subject Classification ID: 55N91
Mathematics Subject Classification ID: 57S17
zbMATH DE Number: 6122386
zbMATH Keywords: equivariant map
zbMATH Keywords: covering dimension
zbMATH Keywords: orthogonal representation
zbMATH Keywords: equivariant \(K\)-theory
reviewed by: Christian Fenske
MaRDI profile type: Publication
full work available at URL: agt.2012.12.2245
OpenAlex ID: W1971803454
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DOI: 10.2140/AGT.2012.12.2245
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description: scientific article; zbMATH DE number 6122386