# Reduced suspension

## Idea

If we take a pointed space $(X,x_0)$, then its reduced suspension $\Sigma X$ is obtained by taking the cylinder $I\times X$ and identifying the subspace $\{0,1\}\times X\cup I\times \{x_0\}$ to a point.

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

Compare the suspension $S X$, where there is no basepoint and only the ends of the cylinder are crushed.

## Definition

For a pointed space $(X,x_0)$,

$\Sigma X = (I\times X)/\{0,1\}\times X\cup I\times \{x_0\}$

This can also be thought of as forming $S^1\wedge X$, the smash product of the circle (based at some point) with $X$:

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

## Properties

### Relation to suspension

For CW-complexes the reduced suspension is weakly homotopy equivalent to the ordinary suspension: $\Sigma X \simeq S X$.

## 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 November 3, 2013 21:33:09 by Urs Schreiber (89.204.139.125)