HOMOGENEITY DEGREE OF HYPERSPACES OF ARCS AND SIMPLE CLOSED CURVES

Abstract

Given a continuum $X$ and $n\in \mathbb{N}$, let $C_n(X)$ (resp. $F_n(X)$) be the hyperspace of nonempty closed sets with at most $n$ components (resp. $n$ points). Given $1\le m\le n$, we consider the quotient space $C_n(X)∕F_m(X)$. The homogeneity degree of $X$, $\text{Hd}(X)$, is the number of orbits of the group of homeomorphisms of $X$. We discuss lower bounds for the homogeneity degree of the hyperspaces $C_n(X)$, $C_n(X)∕F_m(X)$ when $X$ is a finite graph. In particular, we prove that for a finite graph $X$:

• (a) $\text{Hd}(C_n(X)∕F_m(X))=1$ if and only if $X$ is a simple closed curve and $n=m=1$,

• (b) $\text{Hd}(C_n(X)∕F_m(X))=2$ if and only if $X$ is an arc and either $n=m=1$ or $n=2$ and $m\in \{1,2\}$,

• (c) $\text{Hd}(C_n(X)∕F_m(X))=3$ if and only if $X$ is a simple closed curve and $n=m=2$, and

• (d) $\text{Hd}(C_n(X)∕F_m(X))=4$ if and only if $X$ is a simple closed curve, $n=2$ and $m=1$.