Tetrahedra on Deformed and Integral Group Cohomology

Abstract

We show that for every injective continuous map $f:S^2\rightarrow{\Bbb R}^3$ there are four distinct points in the image of $f$ such that the convex hull is a tetrahedron with the property that two opposite edges have the same length and the other four edges are also of equal length. This result represents a partial result for the topological Borsuk problem for ${\Bbb R}^3$. Our proof of the geometrical claim, via Fadell–Husseini index theory, provides an instance where arguments based on group cohomology with integer coefficients yield results that cannot be accessed using only field coefficients.