of the product of X with the unit interval I = [0, 1]. Intuitively, we make X into a cylinder and collapse both ends to two points. One views X as "suspended" between the end points. One can also view the suspension as two cones on X glued together at their base (or as a quotient of a single cone).
Given a continuous map there is a map defined by This makes into a functor from the category of topological spaces into itself. In rough terms S increases the dimension of a space by one: it takes an n-sphere to an (n + 1)-sphere for n ≥ 0.
The space is sometimes called the unreduced, unbased, or free suspension of , to distinguish it from the reduced suspension described below.
The 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.
If X is a pointed space (with basepoint x0), there is a variation of the suspension which is sometimes more useful. The reduced suspension or based suspension ΣX of X is the quotient space:
This is the equivalent to taking SX and collapsing the line (x0 × I) joining the two ends to a single point. The basepoint of ΣX is the equivalence class of (x0, 0).
Σ gives rise to a functor from the category of pointed spaces to itself. An important property of this functor is that it is a left adjoint to the functor taking a (based) space to its loop space . In other words,
naturally, where stands for continuous maps which preserve basepoints. This is not the case for unreduced suspension and free loop space.