CW complexes, the reduced suspension of X is homotopy equivalent to the unbased suspension. Adjunction of reduced suspension and loop space functors Σ gives rise to a functor from the category of pointed spaces to itself. An important property of this functor is that it is left adjoint to the functor taking a pointed space to its loop space . In other words, we have a natural isomorphism where and are pointed spaces and stands for continuous maps that preserve basepoints. This adjunction can be understood geometrically, as follows: arises out of if a pointed circle is attached to every non-basepoint of , and the basepoints of all these circles are identified and glued to the basepoint of . Now, to specify a pointed map from to , we need to give pointed maps from each of these pointed circles to . This is to say we need to associate to each element of a loop in (an element of the loop space ), and the trivial loop should be associated to the basepoint of : this is a pointed map from to . (The continuity of all involved maps needs to be checked.) The adjunction is thus akin to currying, taking maps on cartesian products to their curried form, and is an example of Eckmann–Hilton duality. This adjunction is a special case of the adjunction explained in the article on smash products. Applications The reduced suspension can be used to construct a homomorphism of homotopy groups, to which the Freudenthal suspension theorem applies. In homotopy theory, the phenomena which are preserved under suspension, in a suitable sense, make up stable homotopy theory. Examples Some examples of suspensions are: The suspension of an n-ball is homeomorphic to the (n+1)-ball. Desuspension Desuspension is an operation partially inverse to suspension. See also Double suspension theorem Cone (topology) Join (topology) References Topology Homotopy theory