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

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. Therefore, given an n-dimensional space ${\displaystyle X}$, the desuspension ${\displaystyle \Sigma ^{-1}{X}}$ has dimension n – 1.

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.

See also

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

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. Archived from the original (PDF) on 26 June 2015. Retrieved 25 June 2015.
2. Margolis, Harvey Robert (1983). Spectra and the Steenrod Algebra. North-Holland Mathematical Library. North-Holland. p. 454. ISBN 978-0-444-86516-8. LCCN 83002283.

External links

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

• Topology
• Homotopy theory