Abstract.
Let \(f \colon S_h \to S_g\) be a continuous mapping between orientable closed surfaces of genus h and g and let c denote the constant map \(c \colon S_h \to S_g\) with \(c(S_h) = c\in S_g\). Let \(\varrho(f)\) be the minimal number of roots of f' among all maps f' homotopic to f, i.e. \(\varrho(f) = \min \{|f'^{-1}(c)| : f' \simeq f \colon S_h \to S_g \}\). We prove that
\(\varrho(f) = \max \{\ell(f), d - (d\cdot\chi(S_g) - \chi(S_h) ) \}\) where
\(\ell(f) =[ \pi_1(S_g) : f_{\#}(\pi_1(S_h))]\) and \(\chi\) denotes the Euler characteristic. In addition, certain quadratic equations in free groups closely related to the coincidence problem are solved.
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Received January 26, 1999; in final form November 8, 1999 / Published online February 5, 2001
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Bogatyi, S., Gonçalves, D. & Zieschang, H. The minimal number of roots of surface mappings and quadratic equations in free groups. Math Z 236, 419–452 (2001). https://doi.org/10.1007/s002090100203
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DOI: https://doi.org/10.1007/s002090100203