On equivalences of branched coverings and their action on homology
This paper studies equivalences of stable simple branched coverings of surfaces. We give necessary and sufficient conditions for a pair of homeomorphisms f and g of surfaces M and N respectively to be homologous to homeomorphisms \(\bar f\) and \(\bar g\) which form an equivalence of two prespecified stable simple branched covers \(\psi_ 1\) and \(\psi_ 2\). That is, homeomorphisms \(\bar f\) and \(\bar g\) such that \(\psi_ 2\bar f=\bar g\psi_ 1\) are shown to exist if and only if \(\psi_{2*}f_*=g_*\psi_{1*}\) from \(H_*(M)\) to \(H_*(N).\) The proof relies on a uniqueness theorem of Hamilton and Berstein, Edmonds to restate the problem in terms of self equivalences of certain simple branched covers. Many equivalences of branched covers are constructed, and it is shown that the action on homology of these equivalences generates an appropriate subgroup of the symplectic group.
stable simple branched coverings of surfaces
scientific article; zbMATH DE number 3905462