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2019

Vanishing of all equivariant obstructions and the mapping degree

Avvakumov, Sergey, Kudrya, Sergey

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Suppose that $n\neq p^k$ and $n\neq 2p^k$ for all $k$ and all primes $p$.

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 $X \to {\mathbb R}^n$ whose image avoids the diagonal $\{(x,x\dots,x)\in {\mathbb R}^n|x\in {\mathbb R}\}$.
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.

Vanishing of all equivariant obstructions and the mapping degree · Around

Built on

On certain geometrical properties of closed curves
L. G. Shnirel’man · 1944
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Kneser’s conjecture, chromatic number, and homotopy
L. Lovász · 1978
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On a topological generalization of a theorem of Tverberg
I. Bárány, S. B. Shlosman, and A. Szücs · 1981
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Equilibrium in a discrete exchange economy with money
D. Gale · 1984
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N. Alon · 1987
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