A note on moves and on irregular coverings of S4

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.