Branch point structure of covering maps onto nonorientable surfaces

Abstract

Let f : M → N f:M\, \to \,N be a degree n branched cover onto a compact, connected nonorientable surface with branch points y 1 , y 2 , … , y m {y_1},\,{y_2},\, \ldots ,\,{y_m} in N, and let the multiplicities at points in f − 1 ( y i ) {f^{ - 1}}({y_i}) be μ i 1 , μ i 2 , … , μ i k i {\mu _{i1}},\,{\mu _{i2}},\, \ldots ,\,{\mu _{i{k_i}}} . The branching array of f, designated by B, is the following array of numbers: \[ μ 11 , μ 12 , … , μ 1 k 1 μ 21 , μ 22 , … , μ 2 k 2 ⋮ μ m 1 , μ m 2 , … , μ m k m \begin {gathered} {\mu _{11}},\,{\mu _{12}},\, \ldots ,\,{\mu _{1{k_1}}} {\mu _{21}},\,{\mu _{22\,}}, \ldots ,\,{\mu _{2{k_2}}} \,\,\,\,\,\,\,\,\,\,\,\, \vdots {\mu _{m1}},\,{\mu _{m2}},\, \ldots ,\,{\mu _{m{k_m}}} \end {gathered} \] We show that the numbers in the branching array must always satisfy the following conditions: (1) \[ ∑ { μ i j + 1 | j = 1 , 2 , … , k i } = n \sum {\{ {\mu _{ij}} \,+ \,1|j \,=\, 1,\,2,\, \ldots ,\,{k_i}\} \,=\, n} \] , (2) ∑ { μ i j | i = 1 , 2 , … , m ; j = 1 , 2 , … , k i } \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.