Paths and cylinders
For pointed topological space, then its reduced suspension is equivalently
obtained from the standard cylinder by 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.)
obtained from the bare suspension of and identifying with a single point.
obtained from the reduced cylinder by collapsing the two ends, i.e. the cofiber
the mapping cone in pointed topological spaces formed with respect to the reduced cylinder of the map ;
the smash product , of with the circle (based at some point) with .
Relation to suspension
For CW-complexes that are also pointed, with the point identified with a 0-cell, then their reduced suspension is weakly homotopy equivalent to the ordinary suspension: .
suspensions are H-cogroup objects
Up to homeomorphism, the reduced suspension of the -sphere is the -sphere
See at one-point compactification – Examples – Spheres for details.
Revised on February 13, 2017 03:20:54