A generalization of a result of Dong and Santos-Sturmfels on the Alexander dual of spheres and balls

We prove a generalization of a result by Dong and Santos-Sturmfels about the homotopy type of the Alexander dual of balls and spheres. Our results involve NH-manifolds, which were recently introduced as the non-homogeneous (or non-pure) counterpart of classical polyhedral manifolds. We show that the Alexander dual of an NH-ball is contractible and the Alexander dual of an NH-sphere is homotopy equivalent to a sphere. We also prove that NH-balls and NH-spheres arise naturally as the double duals of standard balls and spheres.