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

José G. Anaya, Alfredo Cano, Enrique Castañeda-Alvarado, Marco A. Castillo-Rubí

Let X be a continuum and n ∈ N, Fn(X) denotes the hyperspace of all subsets of X with at most n points. Given m,n ∈ N with m less than n, we consider SFmn(X) as the quotient space Fn(X)/Fm(X). In this paper we will show that SFmn(-) is a homotopic functor. Thus we will obtain a classification by homotopy. We will study the homotopy type of SF12(X) and we will calculate the Euler characteristic of SFmn(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 ∈ N, with m less than n, if X a continuum such that SFmn(X) is homeomorphic to SFn(X), is X contractible? 

Advanced Studies: Euro-Tbilisi Mathematical Journal, Vol. 16(2) (2023), pp. 29-46