Desuspension
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In topology, a field within mathematics, desuspension is an operation inverse to suspension.
Definition
In general, given an n-dimensional space X {\displaystyle X}, the suspension Σ X {\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 Σ − 1 {\displaystyle \Sigma ^{-1}}, called desuspension. Therefore, given an n-dimensional space X {\displaystyle X}, the desuspension Σ − 1 X {\displaystyle \Sigma ^{-1}{X}} has dimension n – 1.
In general, Σ − 1 Σ X ≠ X {\displaystyle \Sigma ^{-1}\Sigma {X}\neq X}.
Reasons
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.