2023-12-02T12:39:10Zhttps://docta.ucm.es/rest/oai/requestoai:docta.ucm.es:20.500.14352/647062023-08-25T12:44:14Zcom_20.500.14352_14col_20.500.14352_15
Lifting surgeries to branched covering spaces
Montesinos Amilibia, José María
Hilden, Hugh Michael
Long ago J. W. Alexander showed that any closed, orientable, triangulated n-manifold can be expressed as a branched covering of the n-sphere [Bull. Amer. Math. Soc. 26 (1919/20), 370–372; Jbuch 47, 529]. In general, the branch set is not a manifold and no useful information is given about the degree of the branched covering. When n=3, however, he did indicate that the branch set could be arranged to be a link. Much more recently, the first author [Amer. J. Math. 98 (1976), no. 4, 989–997], U. Hirsch [Math. Z. 140 (1974), 203–230] and the second author [Quart. J. Math. Ser. (2) 27 (1976), no. 105, 85–94] showed that when n=3 the branched covering can be constructed to have degree 3 and a knot as branch set. Of course, these branched coverings are highly irregular.
The authors here address similar questions in higher dimensions. Starting with a branched covering Mn→Sn, the authors give some technical, sufficient conditions for a manifold obtained from Mn by a single surgery to be a branched covering of Sn of the same degree and with a branch set easily described in terms of the initial branch set.
The nicest corollary of the general technique is that if Mn→Sn is a branched covering of degree d, then there is a branched covering Mn×Sk→Sn+k of degree d+1. The new branch set is an orientable and/or locally flat submanifold if and only if the original branch set is. In particular, the n-torus is an n-fold branched covering of the n-sphere, branched along a locally flat, orientable submanifold. (For known cohomological reasons, n is the smallest possible degree of such a branched covering.)
2023-06-21T02:02:52Z
2023-06-21T02:02:52Z
2023-06-21T02:02:52Z
1980
journal article
https://hdl.handle.net/20.500.14352/64706
A. Edmonds, Extending a branched covering over a handle (preprint). cf.
I. Berstein and A. Edmonds, On the construction of branched coverings of low-dimensional manifolds, Trans. Amer. Math. Soc. 247 (1979), 87-124.
I. Berstein and A. Edmonds, The degree and branch set of a branched covering, Invent. Math. (to appear).
J. Montesinos, Sobre la conjetura de Poincaré y los recubridores ramificados sobre un nudo, Thesis, Universidad Complutense, Madrid, Spain, 1971.
J. Montesinos, Three-manifolds as 3-fold branched covers of S3, Quart. J. Math. Oxford Ser. (2) 27 (1976), 85-94.
J. Montesinos, 4-manifolds, 3-fold branched covering spaces and ribbons, Trans. Amer. Math. Soc. 245 (1978), 453-467.
0002-9947
10.2307/199815
http://www.ams.org/journals/tran/1980-259-01/S0002-9947-1980-0561830-0/S0002-9947-1980-0561830-0.pdf
http://www.ams.org/
eng
restricted access
American Mathematical Society