Paths and cylinders
If we take a pointed space , then its reduced suspension is obtained by taking the cylinder and identifying the subspace 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 , where there is no basepoint and only the ends of the cylinder are crushed.
For a pointed space ,
This can also be thought of as forming , the smash product of the circle (based at some point) with :
Relation to suspension
For CW-complexes the reduced suspension is weakly homotopy equivalent to the ordinary suspension: .
Up to homeomorphism, the reduced suspension of the -sphere is the -sphere
See at one-point compactification – Examples – Spheres for details.
Revised on November 3, 2013 21:33:09
by Urs Schreiber