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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2502.20505 |
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| _version_ | 1866909516325453824 |
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| author | Jonard-Pérez, Natalia López-Poo, Ananda |
| author_facet | Jonard-Pérez, Natalia López-Poo, Ananda |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_20505 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Equivariant means Jonard-Pérez, Natalia López-Poo, Ananda Geometric Topology 54C05, 54C99, 54C55, 54H15 (Primary) 91B14 (Secondary) 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. |
| title | Equivariant means |
| topic | Geometric Topology 54C05, 54C99, 54C55, 54H15 (Primary) 91B14 (Secondary) |
| url | https://arxiv.org/abs/2502.20505 |