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 \({\mathfrak {S}}_n\), there exists an \({\mathfrak {S}}_n\)-equivariant map \(X \rightarrow {{\mathbb {R}}}^n\) whose image avoids the diagonal \(\{(x,x,\dots ,x)\in {{\mathbb {R}}}^n\mid x\in {{\mathbb {R}}}\}\). 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 \(\mathfrak S_n\)-equivariant maps from the boundary \(\partial \Delta ^{n-1}\) 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.

假设n不是素数，也不是素数的两倍。我们证明，对于具有对称组\（{\ mathfrak {S}} _ n \）自由作用的 任何Hausdorff紧致X，都存在\（{\ mathfrak {S}} _ n \）-等价映射\（X \ rightarrow {{\ mathbb {R}}} ^ n \）的图像避开{{\ mathbb {R}}} ^ n \ mid x中对角线\（\ {（x，x，\ dots，x）\ \ in {{\ mathbb {R}}} \} \）中。以前，对于某些X，此语句的特殊情况通常使用等值障碍理论来证明。这样的计算很困难，并且在第一个（主要）障碍物之后可能变得不可行。我们采用了不同的方法，使我们能够同时证明所有障碍物的消失。证明的关键步骤是根据\（（n-1）\）的边界\（\ partial \ Delta ^ {n-1} \）对\（\ mathfrak S_n \）-等变图的可能度进行分类-对自身简单。对于离散数学和几何学引起的许多问题，例如Kneser猜想，Square Peg猜想，Splitting Necklace问题和拓扑Tverberg猜想等，空间之间存在等变图很重要。对于一个这样的问题，一个令人羡慕的分裂问题的具体实例。