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We show that for every injective continuous map f: S^2 --> R^3 there are fourdistinct points in the image of f such that the convex hull is a tetrahedronwith the property that two opposite edges have the same length and the otherfour edges are also of equal length. This result represents a partial resultfor the topological Borsuk problem for R^3. Our proof of the geometrical claim,via Fadell-Husseini index theory, provides an instance where arguments based ongroup cohomology with integer coefficients yield results that cannot beaccessed using only field coefficients.

Tetrahedra on deformed spheres and integral group cohomology