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Transactions of the American Mathematical Society

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



$ 4$-manifolds, $ 3$-fold covering spaces and ribbons

Author: José María Montesinos
Journal: Trans. Amer. Math. Soc. 245 (1978), 453-467
MSC: Primary 57M10; Secondary 57N15
MathSciNet review: 511423
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Abstract: It is proved that a PL, orientable 4-manifold with a handle presentation composed by 0-, 1-, and 2-handles is an irregular 3-fold covering space of the 4-ball, branched over a 2-manifold of ribbon type. A representation of closed, orientable 4-manifolds, in terms of these 2-manifolds, is given. The structure of 2-fold cyclic, and 3-fold irregular covering spaces branched over ribbon discs is studied and new exotic involutions on $ {S^4}$ are obtained. Closed, orientable 4-manifolds with the 2-handles attached along a strongly invertible link are shown to be 2-fold cyclic branched covering spaces of $ {S^4}$. The conjecture that each closed, orientable 4-manifold is a 4-fold irregular covering space of $ {S^4}$ branched over a 2-manifold is reduced to studying $ \gamma \char93 {S^1} \times {S^2}$ as a nonstandard 4-fold irregular branched covering of $ {S^3}$.

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Keywords: 4-manifolds, 3-manifolds, 3-fold irregular covering spaces, 2-fold cyclic covering spaces, handle presentation, ribbons, exotic involutions, Mazur manifolds, knots, strongly-invertible knots
Article copyright: © Copyright 1978 American Mathematical Society

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