1n-Homogeneity of the 2-nd cones

A space is said to be 1n-homogeneous provided there are exactly n orbits for the action of the group of homeomorphisms of the space onto itself. In this paper, we investigate 1n-homogeneity in suspensions and cones of locally compact, homogeneous and finite dimensional metric spaces, we prove that if X is a solenoid, then the hyperspace of all subcontinua of X, is 13-homogeneous. Moreover, we determine conditions under which the 2-nd cone of a Hausdorff space is 12-homogeneous. Finally, we include a list of open problems related to this topic.