Suspension (topology)

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In topology, the suspension SX of a topological space X is the quotient space:

Suspension of a circle. The original space is in blue, and the collapsed end points are in green.

of the product of X with the unit interval I = [0, 1]. Thus, X is stretched into a cylinder and then both ends are collapsed to 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 homeomorphic to the join where is a discrete space with two points.

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.

Reduced suspension[edit]

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).

One can show that the reduced suspension of X is homeomorphic to the smash product of X with the unit circle S1.

For well-behaved spaces, such as CW complexes, the reduced suspension of X is homotopy equivalent to the ordinary suspension.

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.


Main article: desuspension

Desuspension is an operation inverse to suspension.[1]

See also[edit]


  1. ^ Wolcott, Luke. "Imagining Negative-Dimensional Space" (PDF). Retrieved 2015-06-23.