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 suspension of X is denoted by SX[1] or susp(X).

[2]: 76 There is a variation of the suspension for pointed space, which is called the reduced suspension and denoted by ΣX.

is the result of constructing the cylinder

as two cones on X, glued together at their base.

is a discrete space with two points.

In Homotopy type theory,

be defined as a higher inductive type generated by

[3] In rough terms, S increases the dimension of a space by one: for example, it takes an n-sphere to an (n + 1)-sphere for n ≥ 0.

where square brackets denote equivalence classes.

into a functor from the category of topological spaces to itself.

If X is a pointed space with basepoint x0, 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 (x0 × I ) joining the two ends to a single point.

The basepoint of the pointed space ΣX is taken to be the equivalence class of (x0, 0).

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.

Σ gives rise to a functor from the category of pointed spaces to itself.

taking a pointed space

In other words, we have a natural isomorphism where

stands for continuous maps that preserve basepoints.

This adjunction can be understood geometrically, as follows:

if a pointed circle is attached to every non-basepoint of

(an element of the loop space

), and the trivial loop should be associated to the basepoint of

(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.

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.