A Wecken theorem for \(n\)-valued maps
\textit{B. Jiang} [Am. J. Math. 102, 749--763 (1980; Zbl 0455.55001)] proved that if \(X\) is a compact connected polyhedron different from a 2-manifold then the Nielsen number of an arbitrary self map coincides with the number of fixed points of some map in the homotopy class of the self map. \textit{H. Schirmer} [Fundam. Math. 126, 83--92 (1985; Zbl 0609.55001)] proved the following Theorem: Let \(X\) be a compact manifold (with or without boundary) of dimension greater than or equal to three, and let \(\varphi:X \multimap X\) be an \(n\)-valued map, then there is an \(n\)-valued homotopy between \(\varphi\) and an \(n\)-valued map with \(N(\varphi)\) many fixed points. This result is almost the correspondent of the famous result of Wecken for single valued maps. The paper under review generalizes the result of Schirmer by assuming that \(X\) satisfies a weaker hypothesis. More precisely it shows: Theorem 1 - Let \(X\) be a compact polyhedron without local cut points and such that no connected component is a two-manifold. Let \(\varphi:X \multimap X\) be an \(n\)-valued map. Then there is an \(n\)-valued homotopy between \(\varphi\) and an \(n\)-valued map with \(N(\varphi)\) many fixed points. As one sees, the hypothesis above that \(X\) must satisfy coincides with the weaker hypothesis that \(X\) must satisfy for which these type of results (also called Wecken properties) hold for the single valued case.
scientific article; zbMATH DE number 6115435
scientific article