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

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.

Authors and Affiliations

1. Mechanics and Mathematics Faculty, Moscow State University, Moscow 119899, Russia (e-mail: bogatyi@nw.math.msu.su) , Russia

   Semeon Bogatyi

2. Departamento de Matemática - IME-USP, Caixa Postal 66281 - Agência Cidade de S ao Paulo, 05315-970 - Sao Paulo - SP - Brasil, (e-mail: dlgoncal@ime.usp.br) , Brazil

   Daciberg L. Gonçalves

3. Fakultät für Mathematik, Ruhr - Universität Bochum, 44780 Bochum - Germany, (e-mail: marlene.schwarz@ruhr-uni-bochum.de) , Germany

   Heiner Zieschang

Received January 26, 1999; in final form November 8, 1999 / Published online February 5, 2001

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