# Suspension (topology)

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

$SX = (X \times I)/\{(x_1,0)\sim(x_2,0)\mbox{ and }(x_1,1)\sim(x_2,1) \mbox{ for all } x_1,x_2 \in X\}$
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 $f:X\rightarrow Y,$ there is a map $Sf:SX\rightarrow SY$ defined by $Sf([x,t]):=[f(x),t].$ This makes $S$ 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 $SX$ is homeomorphic to the join $X\star S^0,$ where $S^0$ is a discrete space with two points.

The space $SX$ is sometimes called the unreduced, unbased, or free suspension of $X$, 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

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:

$\Sigma X = (X\times I)/(X\times\{0\}\cup X\times\{1\}\cup \{x_0\}\times I)$.

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.

$\Sigma X \cong S^1 \wedge X$

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 $\Omega$ taking a (based) space $X$ to its loop space $\Omega X$. In other words,

$\operatorname{Maps}_*\left(\Sigma X,Y\right)\cong \operatorname{Maps}_*\left(X,\Omega Y\right)$

naturally, where $\operatorname{Maps}_*\left(X,Y\right)$ stands for continuous maps which preserve basepoints. This is not the case for unreduced suspension and free loop space.