nLab
reduced suspension
Context
Homotopy theory
Background
Variations
Definitions
Paths and cylinders
Homotopy groups
Theorems
Reduced suspension
Idea
For $(X,x) as$ pointed topological space , then its reduced suspension $\Sigma X$ is equivalently

obtained from the standard cylinder $I\times X$ by identifying the subspace $(\{0,1\}\times X) \cup (I\times \{x\})$ to a point.

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

obtained from the bare suspension $S X$ of $X$ and identifying $\{x_0\} \times I$ with a single point.

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 point , 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.

Revised on May 10, 2016 10:41:22
by

Urs Schreiber
(131.220.184.222)