Branch point structure of covering maps onto nonorientable surfaces

Abstract:

Let $f:M \to N$ be a degree n branched cover onto a compact, connected nonorientable surface with branch points ${y_1}, {y_2}, \ldots , {y_m}$ in N, and let the multiplicities at points in ${f^{ - 1}}({y_i})$ be ${\mu _{i1}}, {\mu _{i2}}, \ldots , {\mu _{i{k_i}}}$. The branching array of f, designated by B, is the following array of numbers: \[ \begin {gathered} {\mu _{11}}, {\mu _{12}}, \ldots , {\mu _{1{k_1}}} {\mu _{21}}, {\mu _{22 }}, \ldots , {\mu _{2{k_2}}} \vdots {\mu _{m1}}, {\mu _{m2}}, \ldots , {\mu _{m{k_m}}} \end {gathered} \] We show that the numbers in the branching array must always satisfy the following conditions: (1) \[ \sum {\{ {\mu _{ij}} + 1|j = 1, 2, \ldots , {k_i}\} = n} \], (2) $\sum {\{ {\mu _{ij}}|i = 1, 2, \ldots ,m;j = 1, 2, \ldots ,{k_i}\} }$ is even. Furthermore, if B is any array of numbers satisfying these conditions, and if N is not the projective plane, then there is a branched cover onto N with B as its branching array.