Suspension (topology)

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


 * $$SX = (X \times I)/\{(x_1,0)\sim(x_2,0)\mbox{ and }(x_1,1)\sim(x_2,1) \mbox{ for all } x_1,x_2 \in X\}$$



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 $$f:X\rightarrow Y,$$ 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 increases dimension of a space by one: it takes an n-sphere to an (n + 1)-sphere for n &ge; 0.

Note that $$SX$$ 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 &Sigma;X 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 &times; I) joining the two ends to a single point. The basepoint of &Sigma;X 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.

&Sigma; 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.