Lectures on 3-fold simple coverings and 3-manifolds

Impacto

Montesinos Amilibia, José María (1985) Lectures on 3-fold simple coverings and 3-manifolds. In Combinatorial methods in topology and algebraic geometry. Contemporary Mathematics (44). American Mathematical Society, Providence, pp. 157-177. ISBN 0-8218-5039-3

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Abstract

The author presents various ideas, proofs, constructions and tricks connected with branched coverings of 3-manifolds. After an introductory section on 2-fold branched coverings of S3 the main theme of 3-fold irregular coverings is introduced.
A short proof is given of the Montesinos-Hilden theorem concerning the presentation of a (closed, oriented) 3-manifold as an irregular 3-fold covering of S3. Coloured links, associated with irregular 3-fold coverings, are discussed, and moves on coloured links which do not alter the associated covering.
The last section contains an elegant proof of a theorem of Hilden and the author: Every closed oriented 3-manifold is a simple 3-fold covering of S3 branched over a knot so that the branching cover bounds an embedded disc. A consequence of this is the fact that every such 3-manifold is parallelizable. Finally the following result of H. M. Hilden , M. T. Lozano and the author [Trans. Amer. Math. Soc. 279 (1983), no. 2, 729–735;] is proved: Every closed oriented 3-manifold is the pullback of any 3-fold simple branched covering p:S3→S3 and some smooth map Ω:S3→S3 transversal to the branching set of p. This implies an earlier result of Hilden: the possibility to embed any closed oriented 3-manifold M in S3×D2 so that the composition with the projection in the first factor is a 3-fold simple covering.

Item Type: Book Section 3-manifolds as branched coverings of the 3-sphere; surfaces as branched covers of S 2 ; Dehn surgery; simple covers; coloured links; Poincaré conjecture; parallelizable Sciences > Mathematics > Topology 22069 24 Jun 2013 17:28 12 Dec 2018 15:14