nLab
reduced suspension

Reduced suspension

Idea

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

  • obtained from the standard cylinder I×XI\times X by identifying the subspace ({0,1}×X)(I×{x})(\{0,1\}\times X) \cup (I\times \{x\}) 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 SXS X of XX and identifying {x 0}×I\{x_0\} \times I with a single point.

  • obtained from the reduced cylinder by collapsing the two ends, i.e. the cofiber

    ΣXcofib(XXX(I +)) \Sigma X \simeq cofib(X \vee X \to X \wedge (I_+))
  • the mapping cone in pointed topological spaces formed with respect to the reduced cylinder X(I +)X \wedge (I_+) of the map X*X \to \ast;

  • the smash product S 1XS^1\wedge X, of XX with 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.

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 April 14, 2016 16:12:52 by Urs Schreiber (82.113.98.233)