Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Branch point structure of covering maps onto nonorientable surfaces

Author: Cloyd L. Ezell
Journal: Trans. Amer. Math. Soc. 243 (1978), 123-133
MSC: Primary 55A10
MathSciNet review: 0500900
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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{displaymath}\begin{gathered}{\mu _{11}},\,{\mu _{12}},\, \ldots ,\,{\mu _... ...m1}},\,{\mu _{m2}},\, \ldots ,\,{\mu _{m{k_m}}} \end{gathered} \end{displaymath}

We show that the numbers in the branching array must always satisfy the following conditions: (1)

$\displaystyle \sum {\{ {\mu _{ij}} \,+ \,1\vert j \,=\, 1,\,2,\, \ldots ,\,{k_i}\} \,=\, n} $

, (2) $ \sum {\{ {\mu _{ij}}\vert 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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 55A10

Retrieve articles in all journals with MSC: 55A10

Additional Information

Keywords: Branched cover, branching array, multiplicity of a branched cover, nonorientable surface
Article copyright: © Copyright 1978 American Mathematical Society

American Mathematical Society