Symmetries, topological degree and a theorem of Z. Q. Wang
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.
compact \(G\)-covariant perturbation of the identity
scientific article; zbMATH DE number 720435
scientific article