On the construction of branched coverings of low-dimensional manifolds

Berstein, Israel; Edmonds, Allan L.

doi:10.1090/S0002-9947-1979-0517687-9

On the construction of branched coverings of low-dimensional manifolds
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by Israel Berstein and Allan L. Edmonds

Trans. Amer. Math. Soc. 247 (1979), 87-124

DOI: https://doi.org/10.1090/S0002-9947-1979-0517687-9

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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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