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 f and ḡ which form an equivalence of two
prespecified stable simple branched covers ψ1 and ψ2. That is, homeomorphisms f
and ḡ such that
commutes are shown to exist if and only if ψ2∗f∗ = g∗ψ1∗ 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.
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