Equivariant Nielsen fixed point theory for \(n\)-valued maps
Let \(G\) be a finite group and \(X\) a finite simplicial \(G\)-complex. In this paper, the author introduces a version of Nielsen fixed point theory for \(n\)-valued \(G\)-maps \(\varphi: X \multimap X\). This theory simultaneously extends the equivariant Nielsen fixed point theory developed in \textit{J. Better} [Topol. Methods Nonlinear Anal. 36, No.~1, 179--195 (2010; Zbl 1222.55001)] and the non-equivariant theory for \(n\)-valued maps developed in \textit{H. Schirmer} [Fundam. Math. 124, 207--219 (1984; Zbl 0543.55003)]. The two Nielsen-type numbers introduced in this paper, \(NO_G(\varphi)\) and \(N_G(\varphi)\), are defined in terms of an abstract simplicial complex \(K_\varphi\) associated to the \(n\)-valued \(G\)-map \(\varphi\). Each of \(NO_G(\varphi)\) and \(N_G(\varphi)\) is shown to be \(n\)-valued \(G\)-homotopy invariant and each satisfies a lower bound property: if there is an \(n\)-valued \(G\)-homotopy between \(\varphi\) and \(\psi\), then \(NO_G(\varphi)\leq|\{\theta\mid \theta\text{ is an orbit of fixed points of }\psi\}|\) and \(N_G(\varphi)\leq|\mathrm{Fix}(\psi)|\). In addition, the author establishes a minimality result for these lower bounds for a certain restricted class of finite simplicial \(G\)-complexes. The focus of this paper is in establishing the theory and it does not discuss methods of computing either Nielsen-type number introduced.
scientific article; zbMATH DE number 5718379
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