Homogeneity degree of some symmetric products
  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 $n\in \mathbb{N}$. 
Rodrigo Hern\'andez-Guti\'errez, Ver\'onica Mart\'inez-de-la-Vega
math.GN