Web Definition: Suspension
In topology, the suspension SX of a topological space X is the quotient space : of the product of X with the unit interval I =. Intuitively we make X into a cylinder & 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. it takes an n - sphere to an -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. If X is a pointed space there is a variation of the suspension with is sometimes more useful. This is the equivalent to taking SX & collapsing the line joining the two ends to a single point. The basepoint of Σ X is the equivalence class of. 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 S 1.
Read More »Dictionary Entry: The act of suspending, or the state of being suspended - pendency - as, suspension from a hook.
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