On the existence of \(G\)-equivariant maps
type: Item
instance of: scholarly article
title: On the existence of \(G\)-equivariant maps
zbMATH Open document ID: 1273.55001
author: Francielle R. C. Coelho
author: Denise De Mattos
author: Edivaldo L. dos Santos
published in: Bulletin of the Brazilian Mathematical Society. New Series
publication date: 2013-01-10T00:00:00Z
review text: At times, the non-existence of a certain kind of map has interesting consequences, such as the Borsuk--Ulam Theorem. In this paper the authors show the non--existence of certain \(G\)--equivariant maps between free \(G\)--spaces. Let \(R\) be a PID and \(G\) a compact Lie group. Denote the \(i\)--th Betti number of \(X\) by \(\beta_i(X;R)\) and by \(i(X;R)\) the numerical index, cf. \textit{A. Yu. Volovikov} [Sb. Math. 191, No. 9, 1259--1277 (2000); translation from Mat. Sb. 191, No. 9, 3--22 (2000; Zbl 0987.57016)]. Let \(G\) be a compact Lie group and \(X\), \(Y\) free \(G\)--spaces, Hausdorff, pathwise connected and paracompact. Suppose that for some natural number \(m \geq 1\), \(i(X,R)\geq m+1\) and \(H^{k+1}(Y/G;R) = 0\) for some \(k\), \(1\leq k \leq m\). Then if \(k=m\) and \(\beta_m(X;R) < \beta_{m+1}(BG;R)\), there is no \(G\)--equivariant map \(f:X\to Y\). if \(1 \leq k \leq m\) and \(0 < \beta_{k+1}(BG;R)\), there is no \(G\)--equivariant map \(f:X\to Y\).
Mathematics Subject Classification ID: 55M20
Mathematics Subject Classification ID: 55M35
Mathematics Subject Classification ID: 47H09
zbMATH DE Number: 6124359
zbMATH Keywords: Borsuk-Ulam type results
zbMATH Keywords: non-existence of equivariant maps
reviewed by: Karl Heinz Dovermann
MaRDI profile type: Publication
full work available at URL: s00574-012-0019-x
OpenAlex ID: W1996430936
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DOI: 10.1007/S00574-012-0019-X
description: scientific article; zbMATH DE number 6124359