nLab reduced suspension

Reduced suspension



topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


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topological homotopy theory

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see also algebraic topology



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Reduced suspension


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

SXX×[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.


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


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: ΣXSX\Sigma X \simeq S X.

Cogroup structure

suspensions are H-cogroup objects



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.


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