Mathematics – Geometric Topology
Scientific paper
2007-09-13
Trans. Amer. Math. Soc. 361 (2009), 5885-5920
Mathematics
Geometric Topology
49 pages, 18 figures
Scientific paper
For a branched cover between two closed orientable surfaces, the Riemann-Hurwitz formula relates the Euler characteristics of the surfaces, the total degree of the cover, and the total length of the partitions of the degree given by the local degrees at the preimages of the branching points. A very old problem asks whether a collection of partitions of an integer having the appropriate total length (that we call a candidate cover) always comes from some branched cover. The answer is known to be in the affirmative whenever the candidate base surface is not the 2-sphere, while for the 2-sphere exceptions do occur. A long-standing conjecture however asserts that when the candidate degree is a prime number, a candidate cover is always realizable. In this paper we analyze the question from the point of view of the geometry of 2-orbifolds, and we provide strong supporting evidence for the conjecture. In particular, we exhibit three different sequences of candidate covers, indexed by their degree, such that for each sequence: (1) The degrees giving realizable covers have asymptotically zero density in the naturals; (2) Each prime degree gives a realizable cover.
Pascali Maria Antonietta
Petronio Carlo
No associations
LandOfFree
Surface branched covers and geometric 2-orbifolds does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Surface branched covers and geometric 2-orbifolds, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Surface branched covers and geometric 2-orbifolds will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-214698