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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the construction of branched coverings of low-dimensional manifolds


Authors: Israel Berstein and Allan L. Edmonds
Journal: Trans. Amer. Math. Soc. 247 (1979), 87-124
MSC: Primary 57M10
MathSciNet review: 517687
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Abstract: Several general results are proved concerning the existence and uniqueness of various branched coverings of manifolds in dimensions 2 and 3. The results are applied to give a rather complete account as to which 3-manifolds are branched coverings of $ {S^3}$, $ {S^2}\, \times \,{S^1}$, $ {P^2}\, \times \,{S^1}$, or the nontrivial $ {S^3}$-bundle over $ {S^1}$, and which degrees can be achieved in each case. In particular, it is shown that any closed nonorientable 3-manifold is a branched covering of $ {P^2}\, \times \,{S^1}$ of degree which can be chosen to be at most 6 and with branch set a simple closed curve. This result is applied to show that a closed nonorientable 3-manifold admits an open book decomposition which is induced from such a decomposition of $ {P^2}\, \times \,{S^1}$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1979-0517687-9
PII: S 0002-9947(1979)0517687-9
Keywords: Branched covering, 3-manifold, 2-manifold, nonorientable 3 manifold, open book decomposition, classical knot
Article copyright: © Copyright 1979 American Mathematical Society