Desuspension

In topology, a field within mathematics, desuspension is an operation inverse to suspension.^[1]

In general, given an n-dimensional space ${\displaystyle X}$, the suspension ${\displaystyle \Sigma {X}}$ 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 ${\displaystyle \Sigma ^{-1}}$, called desuspension.^[2] Therefore, given an n-dimensional space ${\displaystyle X}$, the desuspension ${\displaystyle \Sigma ^{-1}{X}}$ has dimension n – 1.

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

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.

• Cone (topology)
• Equidimensionality
• Join (topology)

• Desuspension at an Odd Prime
• When can you desuspend a homotopy cogroup?