Vanishing of All Equivariant Obstructions and the Mapping Degree

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