# 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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## Contents

## Definition

### Long definition

Given a topological space , the suspension of , denoted , is defined as the quotient of (where is the closed unit interval and we use the product topology) by the following two equivalence relations:

and

### Short definition

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 .

## Related constructs

## 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.

### Join

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.