Vanishing of all equivariant obstructions and the mapping degree
Avvakumov, Sergey
Kudrya, Sergey
Computational Theory and Mathematics
Discrete Mathematics and Combinatorics
Geometry and Topology
Theoretical Computer Science
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 Nature
2021
info:eu-repo/semantics/article
doc-type:article
text
http://purl.org/coar/resource_type/c_6501
https://research-explorer.app.ist.ac.at/record/11446
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>
eng
info:eu-repo/semantics/altIdentifier/doi/10.1007/s00454-021-00299-z
info:eu-repo/semantics/altIdentifier/issn/0179-5376
info:eu-repo/semantics/altIdentifier/issn/1432-0444
info:eu-repo/semantics/altIdentifier/arxiv/1910.12628
info:eu-repo/semantics/closedAccess