Suspension (topology)


In topology, 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 ΣX 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 = modulo the equivalence relation generated by
One can view the suspension as two cones on X glued together at their base; it is also homeomorphic to the join where 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 -sphere for n ≥ 0.
Given a continuous map there is a continuous map defined by where square brackets denote equivalence classes. This makes into a functor from the category of topological spaces to itself.

Reduced suspension

If X is a pointed space, there is a variation of the suspension which is sometimes more useful. The reduced suspension or based suspension ΣX of X is the quotient space:
This is the equivalent to taking SX and collapsing the line joining the two ends to a single point. The basepoint of the pointed space ΣX is taken to be the equivalence class of.
One can show that the reduced suspension of X is homeomorphic to the smash product of X with the unit circle S1.
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 taking a pointed space to its loop space. In other words, we have a natural isomorphism
where and are pointed spaces and stands for continuous maps that preserve basepoints. This adjunction can be understood geometrically, as follows. arises out of if a pointed circle is attached to every non-basepoint of , and the basepoints of all these circles are identified and glued to the basepoint of. Now, to specify a pointed map from to, we need to give pointed maps from each of these pointed circles to . This is to say we need to associate to each element of a loop in , and the trivial loop should be associated to the basepoint of : this is a pointed map from to.
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

is an operation partially inverse to suspension.