Minimal number of preimages under maps of surfaces (Q869772)

The authors consider a map \(f:M_1\to M_2\) of closed surfaces and \(c\in M_2\). Call \(A(f)\) the absolute degree of \(f\) (cf., [\textit{D. B. A. Epstein}, Proc. Lond. Math. Soc. 16, 369--383 (1966; Zbl 0148.43103)]) and denote by \(\ell(f)\) the index of \(f_\#(\pi_1(M_1))\) in \(\pi_1(M_2)\). Let \(MR(f):=\min| \bar{f}^{-1}(\{c\})| \) where the minimum is taken over all \(\bar{f}\) which are homotopy equivalent to \(f\). Similarly let \(NR(f)\) denote the Nielsen coincidence number for \(f\) and the constant mapping \(c\). The authors prove that \(MR(f)=NR(f)=0\) provided \(A(f)=0\). If \(A(f)>0\) one gets \(MR(f)=\max\{\ell(f),\chi(M_1)+(1-\chi(M_2))A(f)\}\) and \(NR(f)=\ell(f)\).