The degree of equivariant maps
type: Item
instance of: scholarly article
title: The degree of equivariant maps
zbMATH Open document ID: 1064.57038
author: Yasuhiro Hara
published in: Topology and its Applications
publication date: 2005-03-08T00:00:00Z
review text: The \(G\)-index of a \(G\)-space \(X\) with coefficients in \(K\), \(Ind^G(X,K)\), is the kernel of the cohomology map \(c^*_G: H^*_G(*;K) \to H^*_G(X;K)\), where \(c: X \to *\) is the constant map into the one-point space \(*\). If \(G\) acts freely on \(X\) then \(H^*_G(*;K) \cong H^*(BG;K)\), where \(BG\) is the classifying space for \(G\); if \(\phi: X \to BG\) is a classifying map for \(X\), then under that isomorphism ker \(c^*_G \cong ker \phi*\). If \(X\) and \(Y\) are \(G\)-spaces and \(f: X \to Y\) is an equivariant map then \(Ind^G(Y,K) \subset Ind^G(X,K)\). The component of \(Ind^G(X,K)\) in dimension \(q\) is denoted by \(Ind^G_q(X,K)\). Let \(G\) be a compact Lie group acting freely on compact, connected, smooth manifolds \(M\) and \(N\), both of dimension \(n\). The author studies the degree of \(G\)-maps \(f: M \to N\). In this case \(M/G\) and \(N/G\) are manifolds of dimension \(n-dimG\) and \(H^{n-dimG} (M/G;{\mathbb Z}_2) \cong H^{n-dimG}(N/G;{\mathbb Z}_2) \cong {\mathbb Z}_2\). The main results of the paper are contained in the following theorem. Theorem. Let \(f: M \to N\) be a \(G\)-map. (1) If \(Ind^G_{n-dimG}(N,{\mathbb Z}_2) = Ind^G_{n-dimG}(M,{\mathbb Z}_2) \neq H^{n-dimG}(BG;{\mathbb Z}_2)\) then \(f^*: H^*(N;{\mathbb Z}_2) \to H^*(M;{\mathbb Z}_2)\) is an isomorphism; if both \(M\) and \(N\) are oriented then the degree of \(f\) is odd. (2) If \(Ind^G_{n-dimG}(N,{\mathbb Z}_2) \subset Ind^G_{n-dimG}(M,{\mathbb Z}_2) = H^*(BG;{\mathbb Z}_2)\) (but \(Ind^G_{n-dimG}(N,{\mathbb Z}_2) \neq Ind^G_{n-dimG}(M,{\mathbb Z}_2)\)) then \(f^*: H^*(N/G;{\mathbb Z}_2) \to H^*(M/G;{\mathbb Z}_2)\) is zero; if both \(M\) and \(N\) are oriented then the degree of \(f\) is even. The transfer map (or ``integration along the fibre'' construction) plays a crucial role in the proofs. Analogous results for cohomology mod \(p\), where \(p\) is an odd prime, are also proved. These results are applied to maps of real and complex Stiefel manifolds. For instance, i…
reviewed by: Jan Jaworowski
Mathematics Subject Classification ID: 57R91
Mathematics Subject Classification ID: 58E40
Mathematics Subject Classification ID: 57S15
Mathematics Subject Classification ID: 55R35
Mathematics Subject Classification ID: 55M25
Mathematics Subject Classification ID: 57R19
zbMATH DE Number: 2142283
zbMATH Keywords: degree of a map
zbMATH Keywords: \(G\)-cohomology
zbMATH Keywords: index of a \(G\)-space
zbMATH Keywords: transfer
MaRDI profile type: Publication
full work available at URL: j.topol.2004.08.003
OpenAlex ID: W2067373071
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DOI: 10.1016/J.TOPOL.2004.08.003
description: scientific article; zbMATH DE number 2142283