The minimal number of roots of surface mappings and quadratic equations in free groups

• DOI: 10.1007/s002090100203
• Publication Date: 2016-10-14T05:50:22Z
• Authors: Semeon Bogatyi; Daciberg L. Gonça...; Heiner Zieschang
• Keywords: Euler characteristic; Free group

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