# Desuspension

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

## Definition

In general, given an *n*-dimensional space , the suspension 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 , called desuspension.[2] Therefore, given an *n*-dimensional space , the desuspension has dimension *n* – 1.

Note that in general .

## 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.

## References

- 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. - H. R. Margolis (1983).
*Spectra and the Steenrod Algebra*. North-Holland. p. 454.

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