nLab
reduced suspension
Context
Homotopy theory
homotopy theory

Introductions
Background
Variations
Definitions
Paths and cylinders
Homotopy groups
Theorems
Reduced suspension
Idea
For $(X,x)$ a pointed topological space , then its reduced suspension $\Sigma X$ is equivalently the following

obtained from the standard cylinder $I\times X$ (product topological space with the closed interval $I = [0,1]$ ) by identifying the subspace $(\{0,1\}\times X) \cup (I\times \{x\})$ to a point, i.e. the quotient space (this example )

$(X \times [0,1])/ ( ( X \times \{0,1\} ) \cup ([0,1] \times \{x\}) )$

(Think of crushing the two ends of the cylinder and the line through the base point to a point.)

obtained from the plain suspension

$S X = (X \times [0,1])/( X \times \{0,1\})$

of $X$ by passing to the quotient space which collapses $\{x\} \times I$ to a point (this example )

$\Sigma X \simeq S X / ( \{x\} \times I )$

For the purposes of generalized (Eilenberg-Steenrod) cohomology theory typoically it does not matter whether one evaluates on the standard suspension or the reduced suspension. For example for topological K-theory since since $\{x\} \times I$ is a contractoble closed subspace , then this prop. says that topological vector bundles do not see a difference as long as $X$ is a compact Hausdorff space .

obtained from the reduced cylinder by collapsing the two ends, i.e. the cofiber

$\Sigma X \simeq cofib(X \vee X \to X \wedge (I_+))$

the mapping cone in pointed topological spaces formed with respect to the reduced cylinder $X \wedge (I_+)$ of the map $X \to \ast$ ;

the smash product $S^1\wedge X$ , of $X$ with the circle (based at some point) with $X$ .

$\Sigma X \simeq S^1 \wedge X
\,.$

Properties
Relation to suspension
For CW-complexes $X$ that are also pointed , with the point identified with a 0-cell, then their reduced suspension is weakly homotopy equivalent to the ordinary suspension: $\Sigma X \simeq S X$ .

Cogroup structure
suspensions are H-cogroup objects

Example
Spheres
Up to homeomorphism , the reduced suspension of the $n$ -sphere is the $(n+1)$ -sphere

$\Sigma S^n \simeq S^{n+1}
\,.$

See at one-point compactification – Examples – Spheres for details.