Suspension (topology)

In topology, the device to attach or suspension SX denotes a topological space X is the quotient space

Of the product of X with the unit interval I = [0, 1].

Clearly X is extended only to a " cylinder " whose ends are then combined to form points, and you look at X as between these endpoints " hooked ". Can be the device to attach as two geometric taper on X, which are glued together on their base side, to view. A third possibility is their consideration as the quotient of the topological cone over X, in which the points of the base side are summarized as equivalent.

Device to attach is a functor, which increases the dimension of a space by one:

Reduced device to attach

If X is dotted space ( with base point x0 ), so there is an alternate device to attach of X, which is dotted again: The reduced device to attach ΣX of X is the quotient space:

The construction collapses line ( x0 × I) SX, wherein the ends are combined to form a dot. The base point of ΣX is the equivalence class of ( x0, 0). Σ is Endofunktor in the category dotted spaces.

It can be shown that the reduced device to attach of X homeomorphic to the smash product of X with the unit circle S1 is:

Is the general -fold iterated reduced device to attach essentially the smash product with the sphere:

For CW complexes is homotopy equivalent to the reduced device to attach ordinary.

Properties

• The reduced device to attach is linksadjungiert to form the loop space: Are generates compact, so there is a natural isomorphism

• The functoriality of induced device to attach pictures

• For all