nLab
reduced suspension

Reduced suspension

Idea

For (X,x)(X,x) a pointed topological space, then its reduced suspension ΣX\Sigma X is equivalently the following

Properties

Relation to suspension

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

Cogroup structure

suspensions are H-cogroup objects

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 May 31, 2017 01:42:30 by Urs Schreiber (178.6.236.87)