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

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
.

The reasons to introduce desuspension:

- Desuspension makes the category of spaces a triangulated category.
- If arbitrary coproducts were allowed, desuspension would result in all cohomology functors being representable.