Suspension (topology)

In topology, a branch of mathematics, the suspension of a topological space "X" is intuitively obtained by stretching "X" into a cylinder and then collapsing both end faces to points. One views "X" as "suspended" between these end points. 
The space "SX" is sometimes called the unreduced, unbased, or free suspension of "X", to distinguish it from the reduced suspension Σ"X" of a pointed space described below.
The reduced suspension can be used to construct a homomorphism of homotopy groups, to which the Freudenthal suspension theorem applies. In homotopy theory, the phenomena which are preserved under suspension, in a suitable sense, make up stable homotopy theory.
Definition and properties of suspension.
Given a topological space "X", the suspension of "X" is defined as 
the quotient space of the product of "X" with the unit interval "I" = [0, 1] modulo the equivalence relation formula_2 generated by 
One can view the suspension as two cones on "X" glued together at their base; it is also homeomorphic to the join formula_4 where formula_5 is a discrete space with two points.
In rough terms "S" increases the dimension of a space by one: it takes an "n"-sphere to an ("n" + 1)-sphere for "n" ≥ 0.
Given a continuous map formula_6 there is a continuous map formula_7 defined by formula_8 where square brackets denote equivalence classes. This makes formula_9 into a functor from the category of topological spaces to itself. 
Reduced suspension.
If "X" is a pointed space with basepoint "x"0, there is a variation of the suspension which is sometimes more useful. The reduced suspension or based suspension Σ"X" of "X" is the quotient space:
This is the equivalent to taking "SX" and collapsing the line ("x"0 × "I") joining the two ends to a single point. The basepoint of the pointed space Σ"X" is taken to be the equivalence class of ("x"0, 0).
One can show that the reduced suspension of "X" is homeomorphic to the smash product of "X" with the unit circle "S"1.
For well-behaved spaces, such as CW complexes, the reduced suspension of "X" is homotopy equivalent to the unbased suspension.
Adjunction of reduced suspension and loop space functors.
Σ gives rise to a functor from the category of pointed spaces to itself. An important property of this functor is that it is left adjoint to the functor formula_12 taking a pointed space formula_13 to its loop space formula_14. In other words, we have a natural isomorphism
where formula_13 and formula_17 are pointed spaces and formula_18 stands for continuous maps that preserve basepoints. This adjunction can be understood geometrically, as follows: formula_19 arises out of formula_13 if a pointed circle is attached to every non-basepoint of formula_13, and the basepoints of all these circles are identified and glued to the basepoint of formula_13. Now, to specify a pointed map from formula_19 to formula_17, we need to give pointed maps from each of these pointed circles to formula_17. This is to say we need to associate to each element of formula_13a loop in formula_17 (an element of the loop space formula_28), and the trivial loop should be associated to the basepoint of formula_13: this is a pointed map from formula_13 to formula_28. (The continuity of all involved maps needs to be checked.) 
The adjunction is thus akin to currying, taking maps on cartesian products to their curried form, and is an example of Eckmann–Hilton duality. 
This adjunction is a special case of the adjunction explained in the article on smash products.
Desuspension.
Desuspension is an operation partially inverse to suspension.