Coincidence Wecken homotopies versus Wecken homotopies relative to a fixed homotopy in one of the maps
Given two self-maps which achieve the minimal number of fixed points of all maps in a homotopy class, \textit{H. Schirmer} proposed in [Pac. J. Math. 108, 191--202 (1983; Zbl 0476.54040)] a question: is there a homotopy \(H\) between them such that for each \(t\in I\), \(H_t\) gives a self-map that achieves the minimal number of fixed points? The answer turned out to be negative in general from the examples given by the second author in [Topology Appl. 37, No. 3, 293--297 (1990; Zbl 0715.55003)]. The authors of this paper extend Schirmer's question to the case of coincidence points. Let \((f_1, g_1)\), \((f_2, g_2)\) be two pairs of maps which have the minimal number of coincidence points of all pairs of maps in a given homotopy class of pairs. Two questions may be asked: (1) Can one find a homotopy \((F, G)\) connecting \((f_1, g_1)\) and \((f_2, g_2)\) such that all \((F_t, G_t)\)'s have the minimal number of coincidence points; (2) Can one find a homotopy \((F, G)\) as in (1) if \(F\) is fixed. The main results of this paper concern the equivalence of these two questions for maps between closed surfaces. Some sufficient conditions are obtained. The authors also give many concrete and interesting examples showing that these two question may not be equivalent.
scientific article; zbMATH DE number 5718377
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