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

Given a topological space $X$, the suspension of $X$, denoted $SX$, is defined as the quotient of $X\times I$ by the following two equivalence relations:

$(x_1,0)\sim (x_2,0)$

and

$(x_1,1)\sim (x_2,1)$

Also see:

• 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