# nLab reduced suspension

Reduced suspension

# Reduced suspension

## Idea

For a pointed topological space $(X, x)$, its reduced suspension $\Sigma (X,x)$ is obtained from the plain suspension

$\mathrm{S}X \;\coloneqq\; \frac{ X \,\times\, [-1,\, +1] }{ \left( \begin{array}{c} X \times \{-1\}\mathrlap{,\,} \\ X \times \{+1\} \end{array} \right) }$

of the underlying topological space $X$ by collapsing the meridian through the basepoint $x$ itself to a point — this making $\Sigma(X,x)$ itself a pointed topological space with basepoint the equivalence class of that meridian $mer(x)$:

$\Sigma (X,x) \;\coloneqq\; \frac{ \mathrm{S}X }{ \{x\} \times [-1,1] } \;\;\; \in \;\; Top^{\ast/} \,.$

(Notice that this identifies in particular also the two antipodal “poles” of the plain suspension.)

If $X$ admits the structure of a CW-complex then, under passage to the classical homotopy category of pointed topological spaces (cf. here) this construction models the homotopy pushout of the terminal map $(X,x) \to (\ast,pt)$ along itself, which explains its prevalence in homotopy theory (especially in stable homotopy theory, see also at suspension spectrum).

Moreover, in this case of CW-complexes the underlying space of $\Sigma (X,x)$ (i.e. forgetting its basepoint) is weakly homotopy equivalent to the plain suspension $\mathrm{S} X$ of the underlying space $X$ of $(X,x)$. In this sense, reduced suspension in the context of homotopy theory may be understood as just being plain suspension but with basepoints taken into account.

## Definition

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

• obtained from the standard cylinder $I\times X$ (product topological space with the closed interval $I = [0,1]$) by identifying the subspace $(\{0,1\}\times X) \cup (I\times \{x\})$ to a point, i.e. the quotient space (this example)

$(X \times [0,1])/ ( ( X \times \{0,1\} ) \cup ([0,1] \times \{x\}) )$

(Think of crushing the two ends of the cylinder and the line through the base point to a point.)

• obtained from the plain suspension

$S X = (X \times [0,1])/( X \times \{0\}, X \times \{1\})$

of $X$ by passing to the quotient space which collapses $\{x\} \times I$ to a point (this example)

$\Sigma X \simeq S X / ( \{x\} \times I )$

For the purposes of generalized (Eilenberg-Steenrod) cohomology theory typoically it does not matter whether one evaluates on the standard suspension or the reduced suspension. For example for topological K-theory since $\{x\} \times I$ is a contractible closed subspace, then this prop. says that topological vector bundles do not see a difference as long as $X$ is a compact Hausdorff space.

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

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

• the smash product $S^1\wedge X$, of $X$ with the circle (based at some point) with $X$.

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

## Properties

### Relation to suspension

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

### Cogroup structure

suspensions are H-cogroup objects

## Example

### Spheres

Up to homeomorphism, the reduced suspension of the $n$-sphere is the $(n+1)$-sphere

$\Sigma S^n \simeq S^{n+1} \,.$

See at one-point compactification – Examples – Spheres for details.

## References

Discussion of (reduced) suspension may be found in most introductions to homotopy theory (for discussion of unreduced suspension see also there).

For instance:

Review in the context of stable homotopy theory:

Last revised on January 6, 2023 at 14:28:10. See the history of this page for a list of all contributions to it.