Suspension (topology)

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

${\displaystyle SX=(X\times I)/\{(x_{1},0)\sim (x_{2},0){\mbox{ and }}(x_{1},1)\sim (x_{2},1){\mbox{ for all }}x_{1},x_{2}\in X\}}$
Suspension of a circle. The original space is in blue, and the collapsed end points are in green.

of the product of X with the unit interval I = [0, 1]. Thus, X is stretched into a cylinder and then both ends are collapsed to 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 ${\displaystyle f:X\rightarrow Y,}$ there is a map ${\displaystyle Sf:SX\rightarrow SY}$ defined by ${\displaystyle Sf([x,t]):=[f(x),t].}$ This makes ${\displaystyle S}$ into a functor from the category of topological spaces into itself. 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.

The space ${\displaystyle SX}$ is homeomorphic to the join ${\displaystyle X\star S^{0},}$ where ${\displaystyle S^{0}}$ is a discrete space with two points.

The space ${\displaystyle SX}$ is sometimes called the unreduced, unbased, or free suspension of ${\displaystyle X}$, 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.

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:

${\displaystyle \Sigma X=(X\times I)/(X\times \{0\}\cup X\times \{1\}\cup \{x_{0}\}\times I)}$.

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.

${\displaystyle \Sigma X\cong S^{1}\wedge X}$

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 ${\displaystyle \Omega }$ taking a (based) space ${\displaystyle X}$ to its loop space ${\displaystyle \Omega X}$. In other words,

${\displaystyle \operatorname {Maps} _{*}\left(\Sigma X,Y\right)\cong \operatorname {Maps} _{*}\left(X,\Omega Y\right)}$

naturally, where ${\displaystyle \operatorname {Maps} _{*}\left(X,Y\right)}$ stands for continuous maps which preserve basepoints. This adjunction can be understood as a form of currying, taking maps on cartesian products to their curried form, and is an example of Eckmann–Hilton duality. This is not the case for unreduced suspension and free loop space.

Desuspension

Desuspension is an operation inverse to suspension.[1]