Vanishing of All Equivariant Obstructions and the Mapping Degree

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  • Sergey Avvakumov
  • Sergey Kudrya

Suppose that n is not a prime power and not twice a prime power. We prove that for any Hausdorff compactum X with a free action of the symmetric group Sn, there exists an Sn-equivariant map X→ Rn whose image avoids the diagonal { (x, x, ⋯ , x) ∈ Rn∣ x∈ R}. Previously, the special cases of this statement for certain X were usually proved using the equivartiant obstruction theory. Such calculations are difficult and may become infeasible past the first (primary) obstruction. We take a different approach which allows us to prove the vanishing of all obstructions simultaneously. The essential step in the proof is classifying the possible degrees of Sn-equivariant maps from the boundary ∂Δ n-1 of (n- 1) -simplex to itself. Existence of equivariant maps between spaces is important for many questions arising from discrete mathematics and geometry, such as Kneser’s conjecture, the Square Peg conjecture, the Splitting Necklace problem, and the Topological Tverberg conjecture, etc. We demonstrate the utility of our result applying it to one such question, a specific instance of envy-free division problem.

Original languageEnglish
JournalDiscrete and Computational Geometry
Volume66
Pages (from-to)1202–1216 (
ISSN0179-5376
DOIs
Publication statusPublished - 2021

    Research areas

  • Envy-free divisions, Equivariant mapping degree, Equivariant obstruction

ID: 261614995