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On equivalences of branched coverings and their action on homology

1985; Mathematical Sciences Publishers; Volume: 118; Issue: 1 Linguagem: Inglês

10.2140/pjm.1985.118.133

1945-5844

William Kazez,

Advanced Combinatorial Mathematics

This paper studies equivalences of stable simple branched coverings of surfaces.We give necessary and sufficient conditions for a pair of homeomorphisms / and g of surfaces M and N respectively to be homologous to homeomorphisms / and g which form an equivalence of two prespecified stable simple branched covers ψ x and ψ 2 .That is, homeomorphisms/and g such that M -» Ψll N i M iψ 2 commutes are shown to exist if and only if ψ 2 */* = g*Ψi* from 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.