Symmetries, topological degree and a theorem of Z. Q. Wang
type: Item
instance of: scholarly article
title: Symmetries, topological degree and a theorem of Z. Q. Wang
zbMATH Open document ID: 0819.47076
author: P. J. Rabier
published in: Rocky Mountain Journal of Mathematics
publication date: 1995-08-17T00:00:00Z
review text: The following theorem is proved in the paper: Let \(G\) be a compact Lie group and \(X\) be a real Banach space with a continuous linear \(G\)-action. Let \(\Omega\subset X\) be a \(G\)-invariant bounded open subset and \(f: \overline\Omega\to X\) be a compact \(G\)- covariant perturbation of the identity such that \(0\not\in f(\partial\Omega)\). Then \[ \deg(f,\Omega, 0)= \deg(f^ G, \Omega\cap X^ G, 0)\text{ mod }I^ G, \] where deg denotes the Leray-Schauder degree, \(X^ G\) is the fixed point space of the \(G\)-action, \(f^ G: \overline\Omega\cap X^ G\to X^ G\) is the induced map, and \(I^ G\) is the ideal in \(\mathbb{Z}\) generated by the Euler-Poincaré characteristics \(\chi(G/G_ x)\), \(x\in X\backslash X^ G\), \(G_ x\) being the isotropy subgroup of \(x\) such that \(\text{rank }G_ x= \text{rank } G\). In the Hilbert space setting, the theorem was studied by \textit{Z. Q. Wang} [Chin. Ann. Math., Ser. B 10, No. 4, 520-536 (1989; Zbl 0705.58006)], and in the finite-dimensional setting was essentially given by \textit{R. L. Rubinsztein} [Diss. Math., Warszawa 134, 1-48 (1976; Zbl 0343.57021)]. The paper under review contains a full proof of the theorem (even in the Hilbert case it differs from Wang's proof which contains a few gaps) together with some modifications and corollaries.
Mathematics Subject Classification ID: 47H11
Mathematics Subject Classification ID: 55M25
Mathematics Subject Classification ID: 58D19
Mathematics Subject Classification ID: 58C30
Mathematics Subject Classification ID: 57S25
zbMATH DE Number: 720435
zbMATH Keywords: compact \(G\)-covariant perturbation of the identity
zbMATH Keywords: Leray-Schauder degree
zbMATH Keywords: fixed point space
zbMATH Keywords: Euler-Poincaré characteristics
reviewed by: Roman Srzednicki
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072390
OpenAlex ID: W2045709421
cites work: Q5658847
cites work: Q3704110
cites work: Symmetries, degree, homotopy indices and asymptotically homogeneous problems
cites work: Q3319448
cites work: Q5185568
cites work: Q3988350
cites work: The degree of proper \(C^ 2\) Fredholm mappings: Covariant theory
cites work: Q4102494
cites work: Q3261851
cites work: Q5649693
cites work: On the nonlinear elliptic equations with symmetry
cites work: Q3229429
cites work: New aspects in bifurcation with symmetry
cites work: Topological degree and the theorem of Borsuk for general covariant mappings with applications
cites work: Intrinsic isotropy subgroups of finite groups
cites work: Q3326480
cites work: Symmetries and the calculations of degree
DOI: 10.1216/RMJM/1181072390
description: scientific article; zbMATH DE number 720435