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| In [[topology]], the '''suspension''' ''SX'' of a [[topological space]] ''X'' is the [[quotient space]]:
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| :<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>
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| [[Image:Suspension.svg|right|thumb|Suspension of a circle. The original space is in blue, and the collapsed end points are in green.]]
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| of the [[product topology|product]] of ''X'' with the [[unit interval]] ''I'' = [0, 1]. Intuitively, we make ''X'' into a [[cylinder (geometry)|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 [[cone (topology)|cones]] on ''X'' [[adjunction space|glued together]] at their base (or as a quotient of a single cone).
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| Given a continuous map <math>f:X\rightarrow Y,</math> there is a map <math>Sf:SX\rightarrow SY</math> defined by <math>Sf([x,t]):=[f(x),t].</math> This makes <math>S</math> 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.
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| Note that <math>SX</math> is homeomorphic to the [[join (topology)|join]] <math>X\star S^0,</math> where <math>S^0</math> is a [[discrete space]] with two points.
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| The space <math>SX</math> is sometimes called the '''unreduced''', '''unbased''', or '''free suspension''' of <math>X</math>, to distinguish it from the reduced suspension described below.
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| The suspension can be used to construct a homomorphism of [[homotopy group]]s, 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]].
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| ==Reduced suspension==
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| If ''X'' is a [[pointed space]] (with basepoint ''x''<sub>0</sub>), there is a variation of the suspension which is sometimes more useful. The '''reduced suspension''' or '''based suspension''' Σ''X'' of ''X'' is the quotient space:
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| :<math>\Sigma X = (X\times I)/(X\times\{0\}\cup X\times\{1\}\cup \{x_0\}\times I)</math>.
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| This is the equivalent to taking ''SX'' and collapsing the line (''x''<sub>0</sub> × ''I'') joining the two ends to a single point. The basepoint of Σ''X'' is the [[equivalence class]] of (''x''<sub>0</sub>, 0).
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| One can show that the reduced suspension of ''X'' is [[homeomorphic]] to the [[smash product]] of ''X'' with the [[unit circle]] ''S''<sup>1</sup>.
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| :<math>\Sigma X \cong S^1 \wedge X</math> | |
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| For [[well-behaved]] spaces, such as [[CW complex]]es, the reduced suspension of ''X'' is [[homotopy equivalent]] to the ordinary suspension.
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| 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 <math>\Omega</math> taking a (based) space <math>X</math> to its [[loop space]] <math>\Omega X</math>. In other words,
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| :<math> \operatorname{Maps}_*\left(\Sigma X,Y\right)\cong \operatorname{Maps}_*\left(X,\Omega Y\right)</math>
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| naturally, where <math>\operatorname{Maps}_*\left(X,Y\right)</math> stands for continuous maps which preserve basepoints. This is not the case for unreduced suspension and free loop space.
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| ==See also==
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| *[[Cone (topology)]]
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| *[[Join (topology)]]
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| ==References==
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| *[[Allen Hatcher]], [http://www.math.cornell.edu/~hatcher/AT/ATpage.html ''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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| *{{PlanetMath attribution|id=3984|title=Suspension}}
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| [[Category:Topology]]
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| [[Category:Homotopy theory]]
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