SX≔X×[−1,+1](X×{−1}, X×{+1}) \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 XX by collapsing the meridian through the basepoint xx itself to a point — this making Σ(X,x)\Sigma(X,x) itself a pointed topological space with basepoint the equivalence class of that meridian mer(x)mer(x):

Σ(X,x)≔SX{x}×[−1,1]∈Top */. \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 XX 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)→(*,pt)(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 Σ(X,x)\Sigma (X,x) (i.e. forgetting its basepoint) is weakly homotopy equivalent to the plain suspension SX\mathrm{S} X of the underlying space XX of (X,x)(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)(X,x) a pointed topological space, then its reduced suspension ΣX\Sigma X is equivalently the following:

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

(X×[0,1])/((X×{0,1})∪([0,1]×{x})) (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

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

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

ΣX≃SX/({x}×I) \Sigma X \simeq S X / ( \{x\} \times I )

For the purposes of generalized (Eilenberg-Steenrod) cohomology theory typoically typically it does not matter whether one evaluates on the standard suspension or the reduced suspension. For example fortopological K-theory since {x}×I\{x\} \times I is a contractible closed subspace, then this prop. says that topological vector bundles do not see a difference as long as XX is a compact Hausdorff space.
obtained from the reduced cylinder by collapsing the two ends, i.e. the cofiber

ΣX≃cofib(X∨X→X∧(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 1∧XS^1\wedge X, of XX with the circle (based at some point) with XX.

ΣX≃S 1∧X. \Sigma X \simeq S^1 \wedge X \,.

Properties

Relation to suspension

For CW-complexes XX that are also pointed, with the point identified with a 0-cell, then their reduced suspension is weakly homotopy equivalent to the ordinary suspension: ΣX≃SX\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 n≃S n+1. \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:

Marcelo Aguilar, Samuel Gitler, Carlos Prieto, §2.10 in: Algebraic topology from a homotopical viewpoint, Springer (2008) [[doi:10.1007/b97586](https://link.springer.com/book/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](http://www.ams.org/books/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](https://doi.org/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](http://math.uchicago.edu/~amathew/256y.pdf), pdf]

Last revised on January 1, 2024 at 23:23:07. See the history of this page for a list of all contributions to it.