Homogeneity degree of some symmetric products

TL;DR

This paper calculates the homogeneity degree of symmetric products of manifolds and the interval [0,1], revealing how symmetric operations affect the symmetry properties of these spaces.

Contribution

It determines the homogeneity degree of symmetric products for all manifolds without boundary and for the interval [0,1], providing explicit formulas.

Findings

Homogeneity degree of symmetric products of manifolds is explicitly computed.

Homogeneity degree of symmetric products of [0,1] is determined for all n.

Results reveal how symmetry properties evolve under symmetric product operations.

Abstract

For a metric continuum $X$, we consider the $n^{\tiny\textrm{th}}$-symmetric product $F_{n}(X)$ defined as the hyperspace of all nonempty subsets of $X$ with at most $n$ points. The homogeneity degree $hd(X)$ of a continuum $X$ is the number of orbits for the action of the group of homeomorphisms of $X$ onto itself. In this paper we determine $hd(F_{n}(X))$ for every manifold without boundary and $n\in \mathbb{N}$. We also compute $hd(F_{n}[0,1])$ for all $n\in \mathbb{N}$.