Tetrahedra on deformed spheres and integral group cohomology

Details Tetrahedra on deformed spheres and integral group cohomology Tetrahedra on deformed spheres and integral group cohomology

2008-08-28

arxiv.org/abs/0808.3841v1

Mathematics

Algebraic Topology

8 pages, 5 figures

Scientific paper

We show that for every injective continuous map f: S^2 --> 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 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.

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