On the construction of branched coverings of low-dimensional manifolds

Author:

Berstein Israel,Edmonds Allan L.

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^3} , S 2 × S 1 {S^2}\, \times \,{S^1} , P 2 × S 1 {P^2}\, \times \,{S^1} , or the nontrivial S 3 {S^3} -bundle over S 1 {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 × S 1 {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 × S 1 {P^2}\, \times \,{S^1} .

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

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