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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).
Suspension gives rise to a functor, which in rough terms increases dimension of a space by one: it takes an n-sphere to an (n + 1)-sphere for n ≥ 0.
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 with is sometimes more useful. The reduced 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). Σ then gives rise to a functor from the category of pointed spaces to itself.
One can show that the reduced suspension of X is homeomorphic to the smash product of X with the unit circle S1.
For sufficiently nice spaces (such as CW complexes) the reduced suspension of X is homotopy equivalent to the ordinary suspension.
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