Desuspension

In topology, a field within mathematics, desuspension is an operation inverse to suspension.

Definition
In general, given an n-dimensional space $$X$$, the suspension $$\Sigma{X}$$ has dimension n + 1. Thus, the operation of suspension creates a way of moving up in dimension. In the 1950s, to define a way of moving down, mathematicians introduced an inverse operation $$\Sigma^{-1}$$, called desuspension. Therefore, given an n-dimensional space $$X$$, the desuspension $$\Sigma^{-1}{X}$$ has dimension n – 1.

In general, $$\Sigma^{-1}\Sigma{X}\ne X$$.

Reasons
The reasons to introduce desuspension:
 * 1) Desuspension makes the category of spaces a triangulated category.
 * 2) If arbitrary coproducts were allowed, desuspension would result in all cohomology functors being representable.