TY - JOUR

T1 - Vanishing of All Equivariant Obstructions and the Mapping Degree

AU - Avvakumov, Sergey

AU - Kudrya, Sergey

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

KW - Envy-free divisions

KW - Equivariant mapping degree

KW - Equivariant obstruction

UR - http://www.scopus.com/inward/record.url?scp=85104982406&partnerID=8YFLogxK

U2 - 10.1007/s00454-021-00299-z

DO - 10.1007/s00454-021-00299-z

M3 - Journal article

AN - SCOPUS:85104982406

VL - 66

SP - 1202–1216 (

JO - Discrete & Computational Geometry

JF - Discrete & Computational Geometry

SN - 0179-5376

ER -