# Desuspension

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

## Definition

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.

Note that in general ${\displaystyle \Sigma ^{-1}\Sigma {X}\neq X}$.

## 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

1. Wolcott, Luke; McTernan, Elizabeth (2012). "Imagining Negative-Dimensional Space" (PDF). In Bosch, Robert; McKenna, Douglas; Sarhangi, Reza (eds.). Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture. Phoenix, Arizona, USA: Tessellations Publishing. pp. 637–642. ISBN 978-1-938664-00-7. ISSN 1099-6702. Retrieved 25 June 2015.
2. H. R. Margolis (1983). Spectra and the Steenrod Algebra. North-Holland. p. 454.
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