-homogeneity in symmetric products

A space is -homogeneous provided there are exactly m orbits for the action of the group of homeomorphisms of the space onto itself. In this paper we investigate -homogeneity of the 2-fold symmetric product F2(X) for a continuum X. We prove that if X is a 1-dimensional, compact, connected ANR, then F2(X) is -homogeneous if and only if X is an arc or a simple closed curve. We also show that this characterization does not generalize to higher dimensions. Further, we give necessary and sufficient conditions under which the 2-fold symmetric product of an n-manifold is -homogeneous.