Equilateral triangles on a Jordan curve and a generalization of a theorem of Dold

Abstract

Let γ : S 1 → R 2 be a Jordan curve in the plane. It is a simple topological riddle to determine if there is an equilateral triangle with vertices on γ. By reformulating this question in the paradigm of configuration spaces and test maps, we can solve this riddle using a Borsuk–Ulam type theorem obtained using equivariant methods.