In
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, a field within mathematics, desuspension is an operation
inverse to
suspension
Suspension or suspended may refer to:
Science and engineering
* Suspension (topology), in mathematics
* Suspension (dynamical systems), in mathematics
* Suspension of a ring, in mathematics
* Suspension (chemistry), small solid particles suspend ...
.
Definition
In general, given an ''n''-
dimensional
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
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.
Therefore, given an ''n''-dimensional space
, the desuspension
has dimension ''n'' – 1.
In general,
.
Reasons
The reasons to introduce desuspension:
#Desuspension makes the category of spaces a
triangulated category In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy categ ...
.
#If arbitrary
coproduct
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coprodu ...
s were allowed, desuspension would result in all
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
s being representable.
See also
*
Cone (topology)
In topology, especially algebraic topology, the cone of a topological space X is intuitively obtained by stretching ''X'' into a cylinder and then collapsing one of its end faces to a point. The cone of X is denoted by CX or by \operatorname(X).
...
*
Equidimensionality
In mathematics, especially in topology, equidimensionality is a property of a space that the local dimension is the same everywhere.
Definition (topology)
A topological space ''X'' is said to be equidimensional if for all points ''p'' in ''X'', t ...
*
Join (topology)
In topology, a field of mathematics, the join of two topological spaces A and B, often denoted by A\ast B or A\star B, is a topological space formed by taking the disjoint union of the two spaces, and attaching line segments joining every point in ...
References
{{Reflist
External links
Desuspension at an Odd PrimeWhen can you desuspend a homotopy cogroup?
Topology
Homotopy theory