Equivariant means

TL;DR

The paper analyzes when the existence of an equivariant $n$-mean on a $G$-space $X$ forces $X$ to be a $G$-AR, connecting topological social choice rules with equivariant retract theory. For finite groups, it proves that an equivariant $|G|$-mean (or more generally an equivariant $n$-mean with $|G| mid n$) under standard hypotheses implies that $X$ is a $G$-AR, via a constructive $G$-contraction built from the mean. It also introduces contractive $n$-quasi-means with a bound involving $ olinebreak[4]\lambda ext{ in }(0,1)$ and shows that in a complete metric $G$-space with nonempty $X^{G}$ the quasi-mean yields contractibility, hence a $G$-AR when $X$ is a $G$-ANR. Together, these results extend classical links between mean existence and contractibility to equivariant and quasi-mean frameworks, clarifying how fixed-point and orbit-type structures govern retract properties in $G$-spaces.

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 $p(x_1,\dots, x_n)=p(x_{σ(1)},\dots x_{σ(n)})$ for any permutation $σ$ 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. Juárez-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.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 2.1
• Theorem 2.2
• Definition 2.3
• Theorem 2.4
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3