# Suspension

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.

## properties

- 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 .