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
The suspension of a topological space can be described in the following succinct ways as a quotient space . In other words, we quotient out successively (or simultaneously) by the subspaces and .
In terms of other constructions
Double mapping cylinder
The suspension can be viewed as a case of a double mapping cylinder where and are both one-point spaces and both the maps involved send to the one point.
The suspension can also be viewed as the join of with the 0-sphere .
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 :
The result extends to the zeroth homology if we use reduced homology instead of homology. (Without reduced homology, the formulation becomes more clumsy):
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.