Equivariant means

TL;DR

This paper investigates conditions under which the existence of equivariant means on a topological G-space implies that the space is a G-absolute retract, extending previous work and exploring cases without symmetry constraints.

Contribution

It advances understanding of equivariant means on G-spaces and their relation to G-AR properties, generalizing prior results and removing symmetry assumptions.

Findings

Established conditions linking equivariant means to G-AR spaces.

Extended previous results to broader classes of G-spaces.

Analyzed the impact of removing symmetry conditions on n-means.

Abstract

An $n$-mean (also called a ''topological social choice rule'') on a topological space $X$ is a continuous function $p:X^n\to X$ satisfying $p(x,\dots, x)=x$ for every $x\in X$ and $p(x_1,\dots, x_n)=p(x_{\sigma(1)},\dots x_{\sigma(n)})$ for any permutation $\sigma$ of $\{1,\dots, n\}$. If, in addition, $X$ is a $G$-space and $p$ is equivariant with respect to the diagonal action of $G$ on $X^n$, we say that $p$ is an equivariant $n$-mean. In this paper, we continue the work initiated by H. Ju\'arez-Anguiano about conditions on a $G$-space $X$, under which the existence of an equivariant $n$-mean guarantees that $X$ is a $G$-AR. We also explore this problem when we remove the symmetry condition on the definition of an $n$-mean.