nLab 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 typically 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.

Related concepts

• loop space object, free loop space object,

  • delooping

  • loop space, free loop space, derived loop space

• pointed (∞,1)-category, pointed model category

• suspension object

  • suspension type, reduced suspension type

  • suspension, reduced suspension

    • Freudenthal suspension theorem

    • Spanier-Whitehead category

    • suspension isomorphism

    • suspension of a chain complex

    • Sullivan model of a suspension

References

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

For instance:

• Marcelo Aguilar, Samuel Gitler, Carlos Prieto, §2.10 in: Algebraic topology from a homotopical viewpoint, Springer (2008) [doi:10.1007/b97586]

• Jeffrey Strom, §3.8 and §17 in: Modern classical homotopy theory, Graduate Studies in Mathematics 127, American Mathematical Society (2011) [doi:10.1090/gsm/127]

• Martin Arkowitz, Loop spaces and suspensions, §2.3 in: Introduction to Homotopy Theory, Springer (2011) [doi:10.1007/978-1-4419-7329-0]

Review in the context of stable homotopy theory:

• Michael Hopkins (notes by Akhil Mathew), Lecture 2 of: Spectra and stable homotopy theory (2012) [pdf, pdf]