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A note on moves and on irregular coverings of S4

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Montesinos Amilibia, José María (1985) A note on moves and on irregular coverings of S4. In Combinatorial methods in topology and algebraic geometry. Contemporary Mathematics (44). American Mathematical Society, Providence, pp. 345-349. ISBN 0-8218-5039-3

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Abstract

Given a 3-fold simple (i.e. generic branched) covering p:M→S3, a standard modification (called "move C'' in this paper and due to the author and the reviewer in their theses in 1972) permits one to change the branch set but not the covering manifold M. Hence, given two simple coverings pi:M→S3, i=1,2, of degree three, one may ask whether it is possible to pass from p1 to p2 by a finite sequence of moves C or C−1. Using cobordism techniques developed by the author and the reviewer [cf. the author, Trans. Amer. Math. Soc. 245 (1978/79), 453–467; the reviewer, Manuscripta Math. 29 (1979), no. 1, 1–10;] it is shown that for M=4#(S1×S2) there are simple coverings p1 and p2 which cannot be related by any sequence of moves C, C−1. The proof relies on a result by A. Edmonds [Algebraic and geometric topology, Part 2 (Stanford, Calif., 1976), 13–18, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978;]. The question remains open for M=S3.


Item Type:Book Section
Uncontrolled Keywords:colored knot; symmetric group; 3-fold branched covering of S 3
Subjects:Sciences > Mathematics > Topology
ID Code:22071
Deposited On:24 Jun 2013 17:31
Last Modified:12 Dec 2018 15:14

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