Maps between surfaces

Abstract:

The Uniqueness Conjecture states if $\phi , \psi : M \to N$ are $d$-fold, simple, primitive, branched coverings between closed, connected surfaces, then $\phi$ and $\psi$ are equivalent. The Uniqueness Conjecture is proved in the case that $M$ and $N$ are nonorientable and $N = \mathbf {R}{P^2}$ or Klein bottle. It is also proved in the case that $M$ and $N$ are nonorientable and $d/2 < d\chi (N) - \chi (M)$. As an application it is shown that two $d$-fold, branched coverings $\phi :{M_1} \to N, \psi :{M_2} \to N$ between closed, connected surfaces are branched cobordant.