It has been proved that for certain peculiar decomposition spaces Y of euclidean 3-space E?, YXE1 is homeomorphic to euclidean 4-space, El. In this paper we prove that if a decomposition space Y of E? is generated by a trivial defining sequence whose elements are cubes with handles, and this sequence can be replaced by a toroidal defining sequence, then YXE1 is homeomorphic to E*.