Wecken homotopies (Q485073)

Let \(f,g: X\to Y\) be map pair. The coincidence set of \(f\) and \(g\) is defined to be \(\{x\in X\mid f(x) = g(x) \}\). Nielsen coincidence theory deals with the determination of the number of coincidences under homotopies (of pairs). Clearly, the map pairs having minimal coincidence in the homotopy class of \((f,g)\), which are said to be minimal map pairs, are not unique. One may ask whether two minimal pairs can be connected by a so called Wecken homotopy, being a homotopy such that each level is a minimal map pair. This issue was first noticed by \textit{H. Schirmer} [Pac. J. Math. 108, 191--202 (1983; Zbl 0476.54040)], and a concrete example showing homotopic minimal maps may not be connected by any Wecken homotopy was obtained by \textit{M. R. Kelly} [Topology Appl. 37, No. 3, 293--297 (1990; Zbl 0715.55003)] in the case of fixed points, i.e. \(X=Y\) and \(g\) is the identity. The authors of the paper under review obtain some conditions for the existence of Wecken homotopies, including two cases: (1) \(X=Y=S^2\) for fixed points, i.e. \(g\) is the identity; (2) \(X\) and \(Y\) are surfaces of negative Euler characteristic for roots at some point \(y_0\in Y\), i.e. \(g\) is the constant map to \(y_0\). For the second case, the key point is the following observation: For any Wecken homotopy \(H\), the preimage \(H^{-1}(y_0)\) determines an \(n\)-braid on the surface \(X\), where \(n\) is the minimal number of roots. It is known that a braid can be understood as an automorphism. By using this relation, the authors successfully prove that two maps are connected by a Wecken homotopy if and only if the induced homomorphisms between fundamental groups differ by an automorphism derived from some mentioned braid.