Suspension (topology)

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

$SX=(X\times I)/\{(x_{1},0)\sim (x_{2},0){\mbox{ and }}(x_{1},1)\sim (x_{2},1){\mbox{ for all }}x_{1},x_{2}\in X\}$

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]. 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 $f:X\rightarrow Y,$ there is a map $Sf:SX\rightarrow SY$ defined by $Sf([x,t]):=[f(x),t].$ This makes $S$ 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.

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

The space $SX$ is sometimes called the unreduced, unbased, or free suspension of $X$ , 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:

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

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.

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

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

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

naturally, where $\operatorname {Maps}_{*}\left(X,Y\right)$ stands for continuous maps which preserve basepoints. This is not the case for unreduced suspension and free loop space.

References

Allen Hatcher, Algebraic topology. Cambridge University Presses, Cambridge, 2002. xii+544 pp. ISBN 0-521-79160-X and ISBN 0-521-79540-0
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Suspension (topology)

Reduced suspension

See also

References