[Paper Review] Tetrahedra on deformed spheres and integral group cohomology

Pavle V. M. Blagojević, Günter M. Ziegler|ArXiv.org|Aug 28, 2008

Homotopy and Cohomology in Algebraic Topology2 references3 citations

TL;DR

This paper establishes that every injective continuous map from the 2-sphere to R³ contains four distinct points forming a tetrahedron with D₈ symmetry: four equal-length edges forming a cycle and two equal-length opposite edges. Using Fadell–Husseini index theory with integer coefficients, the authors prove the non-existence of a D₈-equivariant map from a configuration space to a sphere, demonstrating that integer cohomology provides stronger topological obstructions than field coefficients in this case.

ABSTRACT

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.

Motivation & Objective

• To establish the existence of a D₈-symmetric tetrahedron in the image of any injective continuous map f: S² → R³.
• To investigate whether equivariant topology tools, particularly Fadell–Husseini index theory, can resolve geometric problems on deformed spheres.
• To compare the strength of Fadell–Husseini index theory using integer coefficients versus field coefficients (especially F₂) in equivariant topology.
• To demonstrate that integer group cohomology yields non-vanishing obstructions where field coefficients fail, highlighting a novel application of Z-coefficients.

Proposed method

• Define a configuration space Ω = (S²)⁴ \ {(x,y,x,y)} to exclude degenerate quadruples.
• Equip Ω with a D₈-action induced by cyclic permutation (ω) and reflection (j) of the four points.
• Construct a D₈-equivariant test map τ: Ω → U₄ × U₂, where U₄ and U₂ are orthogonal complement subspaces in R⁴ and R².
• Use the test map τ to encode the geometric condition: four cyclic edges equal and two opposite edges equal.
• Prove the non-existence of a D₈-equivariant map Ω → S(U₄ × U₂) via comparison of Serre spectral sequences with Z-coefficients.
• Compute the Fadell–Husseini index of Ω and S(U₄ × U₂) using group cohomology of the subgroup ℤ₄ < D₈, showing Index_Z(Ω) = ⟨U³⟩ and Index_Z(S(U₄×U₂)) = ⟨2U²⟩, with 2U² ∉ ⟨U³⟩.

Experimental results

Research questions

• RQ1 Does every continuous injective image of S² in R³ contain four points forming a D₈-symmetric tetrahedron?
• RQ2 Can Fadell–Husseini index theory with integer coefficients detect equivariant obstructions that field coefficients cannot?
• RQ3 Is the topological Borsuk problem for R³ partially resolved by this geometric configuration?
• RQ4 Why does the Fadell–Husseini index with F₂-coefficients fail to obstruct the existence of such a tetrahedron?
• RQ5 To what extent can the configuration space method used here be adapted to the square peg problem?

Key findings

• The paper proves that for every injective continuous map f: S² → R³, there exist four distinct points in f(S²) forming a tetrahedron with four equal cyclic edges and two equal opposite edges.
• The Fadell–Husseini index with Z-coefficients of the configuration space Ω is ⟨U³⟩, a non-trivial ideal in the group cohomology of ℤ₄.
• The Fadell–Husseini index of the target sphere S(U₄ × U₂) with Z-coefficients is ⟨2U²⟩, and since 2U² is not in ⟨U³⟩, no D₈-equivariant map Ω → S(U₄ × U₂) exists.
• With F₂-coefficients, the index of S(U₄ × U₂) vanishes (is trivial), while the index of Ω is ⟨eu², u³⟩, so no obstruction is detected, showing F₂-coefficients are insufficient.
• The proof shows that integer group cohomology provides stronger topological obstructions than field coefficients in this setting, a key methodological contribution.
• The result is robust: the ambient space R³ can be replaced by any metric space (M,d), as the proof depends only on metric structure and continuity.