Desuspension

Definition

In general, given an n-dimensional space X , the suspension Σ 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 Σ − 1 , called desuspension. Therefore, given an n-dimensional space X , the desuspension Σ − 1 X has dimension n – 1.

Note that in general Σ − 1 Σ X ≠ X ≠ Σ Σ − 1 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.