Suspension (topology)

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In topology, the suspension SX of a topological space X is the quotient space:

<math>SX = (X \times I)/\{(x_1,0)\sim(x_2,0)\mbox{ and }(x_1,1)\sim(x_2,1) \mbox{ for all } x_1,x_2 \in X\}</math>

of the product of X with the unit interval I = [0, 1]. Intuitively we make X into a cylinder and collapse both ends to two points. One views X as "suspended" between the end points. One can also view the suspension as two cones on X glued together at their base (or as a quotient of a single cone).

Given a continuous map <math>f:X\rightarrow Y,</math> there is a map <math>Sf:SX\rightarrow SY</math> defined by <math>Sf([x,t]):=[f(x),t].</math> This makes <math>S</math> into a functor from the category of topological spaces into itself. In rough terms increases dimension of a space by one: it takes an n-sphere to an (n + 1)-sphere for n ≥ 0.

Note that <math>SX</math> is homeomorphic to the join <math>X\star S^0,</math> where <math>S^0</math> is a discrete space with two points.

The space <math>SX</math> is sometimes called the unreduced, unbased or free suspension of <math>X</math>, to distinguish it from the reduced suspension described below.

The 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.

[edit] 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:

<math>\Sigma X = (X\times I)/(X\times\{0\}\cup X\times\{1\}\cup \{x_0\}\times I)</math>.

This is the equivalent to taking SX and collapsing the line (x₀ × I) joining the two ends to a single point. The basepoint of ΣX is 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.

<math>\Sigma X \cong S^1 \wedge X</math>

For well-behaved spaces, such as CW complexes, the reduced suspension of X is homotopy equivalent to the ordinary suspension.

Σ gives rise to a functor from the category of pointed spaces to itself. An important property of this functor is that it is a left adjoint to the functor <math>\Omega</math> taking a (based) space <math>X</math> to its loop space <math>\Omega X</math>. In other words,

<math> \operatorname{Maps}_*\left(\Sigma X,Y\right)\cong \operatorname{Maps}_*\left(X,\Omega Y\right)</math>

naturally, where <math>\operatorname{Maps}_*\left(X,Y\right)</math> stands for continuous maps which preserve basepoints.