Connectivity and homology of finite subset spaces of cardinality at most four

The aim of this paper is to study topological features of ; the finite subset space of cardinality at most four associated to a topological space X. We determine its connectivity in terms of the connectivity of X, hence answering a conjecture of Christopher Tuffley in that case. We compute the Euler characteristic of for any topological space X. We provide complete homology computations and determine the rational homotopy type of for any d-dimensional sphere.