# Reduced suspension

## Idea

If we take a pointed space $\left(X,{x}_{0}\right)$, then its reduced suspension $\Sigma X$ is obtained by taking the cylinder $I×X$ and identifying the subspace $\left\{0,1\right\}×X\cup I×\left\{{x}_{0}\right\}$ 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 $SX$, where there is no basepoint and only the ends of the cylinder are crushed.

## Definition

For a pointed space $\left(X,{x}_{0}\right)$,

$\Sigma X=\left(I×X\right)/\left\{0,1\right\}×X\cup I×\left\{{x}_{0}\right\}$\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 $X$:

$\Sigma X\simeq {S}^{1}\wedge X$\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: $\Sigma X\simeq SX$.

## Example

### Spheres

Up to homeomorphism, the reduced suspension of the $n$-sphere is the $\left(n+1\right)$-sphere

$\Sigma {S}^{n}\simeq {S}^{n+1}\phantom{\rule{thinmathspace}{0ex}}.$\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)