In topology, a branch of mathematics, the suspension of a topological spaceX 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. The "usual" suspension SX is sometimes called the unreduced suspension, unbased suspension, or free suspension of X, to distinguish it from ΣX.
Free suspension
The (free) suspension of a topological space can be defined in several ways.
1. is the quotient space In other words, it can be constructed as follows:
Consider the entire set as a single point ("glue" all its points together).
Consider the entire set as a single point ("glue" all its points together).
2. Another way to write this is:
Where are two points, and for each i in {0,1}, is the projection to the point (a function that maps everything to ). That means, the suspension is the result of constructing the cylinder, and then attaching it by its faces, and , to the points along the projections .
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 circleS1.
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.)
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.