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
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 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.
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).
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 (an element of the loop space ), and the trivial loop should be associated to the basepoint of : this is a pointed map from to . (The continuity of all involved maps needs to be checked.)
This adjunction is a special case of the adjunction explained in the article on smash products.
- Wolcott, Luke. "Imagining Negative-Dimensional Space" (PDF). forthelukeofmath.com. Retrieved 2015-06-23.