Suspension

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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

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.