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Desuspension

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

Contents

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.

References

Desuspension Wikipedia


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