On the (n,m)-fold symmetric product suspensions of a finite graph

Abstract

Let $X$ be a continuum and $n\in \mathbb{N}$, $F_n(X)$ denotes the hyperspace of all subsets of $X$ with at most $n$ points. Given $m,n\in \mathbb{N}$ with , we consider $S{F}_m^n(X)$ as the quotient space $F_n(X)/F_m(X)$. In this paper we will show that $S{F}_m^n(-)$ is a homotopic functor. Thus we will obtain a classification by homotopy. We will study the homotopy type of $S{F}_1^2(X)$ and we will calculate the Euler characteristic of $S{F}_m^n(X)$, when $X$ is a finite graph. Finally, we define a polynomial associated with a finite graph to give particular solutions to the problem: Given $m,n\in \mathbb{N}$, with , if $X$ a continuum such that $S{F}_m^n(X)$ is homeomorphic to $F_n(X)$, is $X$ contractible?