“Suspension (topology)”的意思、由来-开放百科全书

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

the quotient space of the product of X with the unit interval I = [0, 1] 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 (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.

Reduced suspension

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.

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