Suspension (topology)

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

Related Topics:
Topology - Topological space - Quotient space

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:SX = (X imes 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}

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of the product of X with the unit interval I = . 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).

Related Topics:
Product - Unit interval - Cylinder - Cones - Glued together

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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.

Related Topics:
Functor - Sphere

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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.

Related Topics:
Homotopy group - Freudenthal suspension theorem - Homotopy theory - Stable homotopy theory

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