nLab
reduced suspension

Reduced suspension

Idea

If we take a pointed space (X,x 0)(X,x_0), then its reduced suspension ΣX\Sigma X is obtained by taking the cylinder I×XI\times X and identifying the subspace {0,1}×XI×{x 0}\{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 SXS X, where there is no basepoint and only the ends of the cylinder are crushed.

Definition

For a pointed space (X,x 0)(X,x_0),

ΣX=(I×X)/{0,1}×XI×{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 1XS^1\wedge X, the smash product of the circle (based at some point) with XX:

ΣXS 1X \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: ΣXSX\Sigma X \simeq S X.

Example

Spheres

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

ΣS nS n+1. \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)