(n,m)-fold covers of spheres

Bárány, Imre and Fabila-Monroy, R. and Vogtenhuber, B. (2015) (n,m)-fold covers of spheres. Proceedings of the Steklov Institute of Mathematics, 288 (1). pp. 203-208. ISSN 0081-5438

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Abstract

A well-known consequence of the Borsuk-Ulam theorem is that if the d-dimensional sphere Sd is covered with less than d + 2 open sets, then there is a set containing a pair of antipodal points. In this paper we provide lower and upper bounds on the minimum number of open sets, not containing a pair of antipodal points, needed to cover the d-dimensional sphere n times, with the additional property that the northern hemisphere is covered m > n times. We prove that if the open northern hemisphere is to be covered m times, then at least ⌈(d − 1)/2⌉ + n + m and at most d + n + m sets are needed. For the case of n = 1 and d ≥ 2, this number is equal to d + 2 if m ≤ ⌊d/2⌋ + 1 and equal to ⌊(d − 1)/2⌋ + 2 + m if m > ⌊d/2⌋ + 1. If the closed northern hemisphere is to be covered m times, then d + 2m − 1 sets are needed; this number is also sufficient. We also present results on a related problem of independent interest. We prove that if Sd is covered n times with open sets not containing a pair of antipodal points, then there exists a point that is covered at least ⌈d/2⌉ + n times. Furthermore, we show that there are covers in which no point is covered more than n + d times. © 2015, Pleiades Publishing, Ltd.

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