Desuspension

Template:Short description In topology, a field within mathematics, desuspension is an operation inverse to suspension.

Definition

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

In general, ${\mathrm{\Sigma }}^{-1}\mathrm{\Sigma }X\ne X$.

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.

See also

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

External links

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