article
Vanishing of all equivariant obstructions and the mapping degree
published
yes
Sergey
Avvakumov
author 3827DAC8-F248-11E8-B48F-1D18A9856A87
Sergey
Kudrya
author ecf01965-d252-11ea-95a5-8ada5f6c6a67
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 ∂Δn−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.
Springer Nature2021
eng
Computational Theory and MathematicsDiscrete Mathematics and CombinatoricsGeometry and TopologyTheoretical Computer Science
Discrete & Computational Geometry
0179-5376
1432-0444
1910.1262810.1007/s00454-021-00299-z
6631202-1216
https://research-explorer.app.ist.ac.at/record/8182
yes
Avvakumov, Sergey, and Sergey Kudrya. “Vanishing of All Equivariant Obstructions and the Mapping Degree.” <i>Discrete & Computational Geometry</i>, vol. 66, no. 3, Springer Nature, 2021, pp. 1202–16, doi:<a href="https://doi.org/10.1007/s00454-021-00299-z">10.1007/s00454-021-00299-z</a>.
Avvakumov S, Kudrya S. 2021. Vanishing of all equivariant obstructions and the mapping degree. Discrete & Computational Geometry. 66(3), 1202–1216.
S. Avvakumov and S. Kudrya, “Vanishing of all equivariant obstructions and the mapping degree,” <i>Discrete & Computational Geometry</i>, vol. 66, no. 3. Springer Nature, pp. 1202–1216, 2021.
Avvakumov S, Kudrya S. Vanishing of all equivariant obstructions and the mapping degree. <i>Discrete & Computational Geometry</i>. 2021;66(3):1202-1216. doi:<a href="https://doi.org/10.1007/s00454-021-00299-z">10.1007/s00454-021-00299-z</a>
Avvakumov, Sergey, and Sergey Kudrya. “Vanishing of All Equivariant Obstructions and the Mapping Degree.” <i>Discrete & Computational Geometry</i>. Springer Nature, 2021. <a href="https://doi.org/10.1007/s00454-021-00299-z">https://doi.org/10.1007/s00454-021-00299-z</a>.
S. Avvakumov, S. Kudrya, Discrete & Computational Geometry 66 (2021) 1202–1216.
Avvakumov, S., & Kudrya, S. (2021). Vanishing of all equivariant obstructions and the mapping degree. <i>Discrete & Computational Geometry</i>. Springer Nature. <a href="https://doi.org/10.1007/s00454-021-00299-z">https://doi.org/10.1007/s00454-021-00299-z</a>
114462022-06-17T08:45:15Z2022-06-17T08:59:13Z