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Functions defined on spheres - remarks on a paper by K. Zarankiewicz

Krasinkiewicz J.

Bulletin of the Polish Academy of Sciences. Mathematics

2001

Vol. 49, no 3

229-242

First we show that for any mapping f : [S^2] --> R there exist two antipodal points p, -p [belongs to S^2] and a continuum C [is a subset of S^2] connecting them such that f is constant on C (Corollary 3.2)^(1); if f is equivariant (with respect to the canonical involutions) then C can be chosen symmetric (Corollary 3.2 or Lemma 5.4). This, combined with a result about the equalization of mappings (Lemma 4.1), slightly improves a classical result of Livesay [8] and Zarankiewicz [12], leading to the notion of functional rectangle of a mapping [S^2] --> R. Such "rectangles" are symmetric continua in [S^2]. Next wre prove that for any mapping f : S^n --> R, n [is greater than or equal to 2], the Borsuk-Ulam set of f, A(f) = {x [belongs to S^n] : f(x) = f(-x)}, contains a unique symmetric component D; it separates [S^n] between antipodal points x, -x for each x [belongs to S^n]\D (Theorem 5.6). Each functional rectangle of f lies in the symmetric component (Corollary 5.7). Some open problems have been posed in the final section.