nLab
reduced suspension

Idea

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

Definition

For a pointed space $(X,x_0)$,

$$\Sigma X=(I\times X)/\{0,1\}\times X\cup I\times \{x_0\}$$\Sigma X = (I\times X)/\{0,1\}\times X\cup I\times \{x_0\}

This can also be thought of as forming $S^1\wedge X$, the smash product of the circle (based at some point) with $X$:

$$\Sigma X\simeq S^1\wedge X$$\Sigma X \simeq S^1 \wedge X

For CW-complexes the reduced suspension is weakly homotopy equivalent to the ordinary suspension: $\Sigma X\simeq SX$.

Related concepts

• loop space object, free loop space object,

  • delooping

  • loop space, free loop space, derived loop space

• suspension object

  • suspension, reduced suspension