Desuspension
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Short description: Mathematical operation inverse to suspension
In topology, a field within mathematics, desuspension is an operation inverse to suspension.[1]
Definition
In general, given an n-dimensional space [math]\displaystyle{ X }[/math], the suspension [math]\displaystyle{ \Sigma{X} }[/math] 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 [math]\displaystyle{ \Sigma^{-1} }[/math], called desuspension.[2] Therefore, given an n-dimensional space [math]\displaystyle{ X }[/math], the desuspension [math]\displaystyle{ \Sigma^{-1}{X} }[/math] has dimension n – 1.
In general, [math]\displaystyle{ \Sigma^{-1}\Sigma{X}\ne X }[/math].
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.
See also
References
- ↑ Wolcott, Luke; McTernan, Elizabeth (2012). "Imagining Negative-Dimensional Space". Phoenix, Arizona, USA: Tessellations Publishing. pp. 637–642. ISBN 978-1-938664-00-7. http://bridgesmathart.org/2012/cdrom/proceedings/65/paper_65.pdf. Retrieved 25 June 2015.
- ↑ Margolis, Harvey Robert (1983). Spectra and the Steenrod Algebra. North-Holland Mathematical Library. North-Holland. p. 454. ISBN 978-0-444-86516-8.
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