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.
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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 $X\times \{1\}$.

Related constructs

• Suspension functor
• Reduced suspension

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 $X$:

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

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

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

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.