Suspension

This article describes a construct that involves some variant of taking a product of a topological space with the unit interval and then making some identifications, typically at the endpoints, based on some specific maps.
View more such constructs

Definition

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 closed unit interval and we use the 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 $X \times \{ 1 \}$.

In terms of other constructions

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 $S^0$.

Relation between a space and its suspension

Homology for suspension

Further information: 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.