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Relative preimage problem. (English. Russian original) Zbl 1118.55002
Math. Notes 80, No. 2, 272-283 (2006); translation from Mat. Zametki 80, No. 2, 281-295 (2006).
This paper is a relative version of [R. Dobrenko and Z. Kucharski, Fundam. Math. 134, No. 1, 1–14 (1990; Zbl 0719.55002)]. Given a relative map \(f\colon (X,X_0)\to (Y, Y_0)\) and a subset \(B\) of \(Y\), the author of this paper discusses the estimation of the minimal number \(MP_{\text{rel}}(f, B)=\min _g | g^{-1}(B)| \) of preimage points at \(B\), where \(g: (X,X_0)\to (Y, Y_0)\) runs over all maps in the relative homotopy class of \(f\). Some relative Nielsen numbers are introduced according to the standard technique in ordinary relative Nielsen fixed point theory [H. Schirmer, Pac. J. Math. 122, 459–473 (1986; Zbl 0553.55001)]. Hence, a lower bound \(N_{\text{rel}, a, a}(f, f_0)\) for \(MP_{\text{rel}}(f, B)\) is obtained. It is shown that this lower bound can be realized under suitable assumptions on \(X, X_0, Y, Y_0, B\), i.e. there exists a map \(g: (X,X_0)\to (Y, Y_0)\) relatively homotopic to \(f\) with \(| g^{-1}(B)| = N_{\text{rel}, a, a}(f, f_0)\) for any given map \(f: (X,X_0)\to (Y, Y_0)\).
Furthermore, the above treatment can be applied to consider the common preimage set \(\{x\in X \mid f_1(x)= \cdots =f_r(x)\in B\}\) for finitely many relative maps \(f_j\colon (X,X_0)\to (Y, Y, Y_0)\), \(j=1,2,\ldots, r\), because the common preimage set is equal to the preimage of the map \(\Delta\circ(f_1, f_2, \ldots, f_r)\) at the set \(\Delta(B)\), in which \(\Delta\colon Y \to \Pi_{j=1}^r Y\) is the diagonal map. It is mentioned by the author that this work formulates the relative coincidence point problem [J. Jezierski, Fundam. Math. 149, No. 1, 1–18 (1996; Zbl 0846.55003)] when \(r=2\), and therefore the relative Nielsen fixed problem.

55M20 Fixed points and coincidences in algebraic topology
54H25 Fixed-point and coincidence theorems (topological aspects)
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