Suspension

Long definition
Given a topological space $$X$$, the suspension of $$X$$, denoted $$SX$$, is defined as the quotient of $$X \times I$$ (where $$I$$ is the defining ingredient::closed unit interval and we use the defining ingredient::product topology) by the following two equivalence relations:

$$\! (x_1,0) \sim (x_2,0), \forall \ x_1,x_2 \in X$$

and

$$\! (x_1,1) \sim (x_2,1) \ \forall \ x_1,x_2 \in X$$

Short definition
The suspension $$SX$$ of a topological space $$X$$ can be described in the following succinct ways as a quotient space $$SX = (X \times [0,1]/(X \times \{ 0 \}))/(X \times \{ 1 \})$$. In other words, we quotient out successively (or simultaneously) by the subspaces $$X \times \{ 0 \}$$ and $$X \times \{ 1 \}$$.

Related constructs

 * Suspension functor
 * Reduced suspension

Double mapping cylinder
The suspension can be viewed as a case of a double mapping cylinder where $$Y$$ and $$Z$$ are both one-point spaces and both the maps involved send $$X$$ to the one point.

Join
The suspension can also be viewed as the join of $$X$$ with the 0-sphere $$S^0$$.

Homology for suspension
Taking the suspension shifts the homology groups. Specifically, for any topological space $$X$$:

$$H_{k + 1}(SX) \cong H_k(X), \qquad k \ge 1$$

The result extends to the zeroth homology if we use reduced homology instead of homology. (Without reduced homology, the formulation becomes more clumsy):

$$\tilde{H}_{k+1}(SX) \cong \tilde{H}_k(X), \qquad k \ge 0$$

This result is an easy application of the Mayer-Vietoris homology sequence, and is similar to the application of the Seifert-van Kampen theorem to show that suspension of path-connected space is simply connected.