desuspension

In general, given an n-dimensional space $X$, the suspension $\backslash Sigma\{X\}$ 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 $\backslash Sigma^\{-1\}$, called desuspension.{{cite book|author=Margolis|first=Harvey Robert|title=Spectra and the Steenrod Algebra|publisher=North-Holland|year=1983|isbn=978-0-444-86516-8|series=North-Holland Mathematical Library|page=454|lccn=83002283}} Therefore, given an n-dimensional space $X$, the desuspension $\backslash Sigma^\{-1\}\{X\}$ has dimension n – 1.

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

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)

{{Reflist}}

• [https://static-content.springer.com/lookinside/chp%3A10.1007%2FBFb0075577/000.png Desuspension at an Odd Prime]
• [https://mathoverflow.net/q/4117 When can you desuspend a homotopy cogroup?]

Category:Topology

Category:Homotopy theory