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Suspension of a circle. The original space is blue, the collapsed endpoints are green.

In topology , the suspension or suspension SX of a topological space X denotes the quotient space

of the product of X with the unit interval I = [0, 1].

Clearly, X is first expanded into a " cylinder "  , the ends of which are then combined into points, and X is viewed as "suspended" between these end points. The suspension can also be viewed as two geometric cones over X , which are glued together on their base. A third possibility is to consider it as the quotient of the topological cone over X , in which the points on the base side are summarized as equivalent.

Suspension is a functor that increases the dimension of a room by one:

Reduced suspension

If there is a dotted space (with a base point ), there is a modified suspension of , which is again dotted: The reduced suspension of is the quotient space:


The construction collapses the straight line ( x 0 × I ) in SX , whereby the ends are combined to one point. The base point of Σ X is the equivalence class of ( x 0 , 0). Σ is an endo-functor in the category of dotted spaces.

One can show that the reduced suspension of X is homeomorphic to the smash product of X with the unit circle S 1 :


more generally, the -fold iterated reduced suspension is essentially the smash product with the - sphere :


For CW complexes , the reduced suspension is homotopy equivalent to the usual one.


  • The reduced suspension is left adjoint to the formation of the loop space : If they are compact , there is a natural isomorphism
In particular,
  • The functoriality of the suspension induces images
between homotopy groups . The Freudenthalsche Einhängungssatz states that these figures for - contiguous spaces in the field isomorphisms and epimorphisms are. The direct Limes
above these maps is the -th stable homotopy group of . In particular , the inductive system is essentially constant for, i.e. H.
Because of this, the groups are simply called stable homotopy groups of the spheres .
  • For all true
If you use reduced homology or reduced cohomology, it even applies to all
This suspension isomorphism (or suspension isomorphism ) also applies to all generalized cohomology theories .

See also