too. Reduced join Given basepointed CW complexes and , the "reduced join" is homeomorphic to the reduced suspensionof the smash product. Consequently, since is contractible, there is a homotopy equivalence This equivalence establishes the isomorphism . Homotopical connectivity Given two triangulable spaces , the homotopical connectivity () of their join is at least the sum of connectivities of its parts: . As an example, let be a set of two disconnected points. There is a 1-dimensional hole between the points, so . The join is a square, which is homeomorphic to a circle that has a 2-dimensional hole, so . The join of this square with a third copy of is a octahedron, which is homeomorphic to , whose hole is 3-dimensional. In general, the join of n copies of is homeomorphic to and . See also Desuspension References Hatcher, Allen, Algebraic topology. Cambridge University Press, Cambridge, 2002. xii+544 pp. and Brown, Ronald, Topology and Groupoids Section 5.7 Joins. Algebraic topology Operations on structures