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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\}

of the Product of ''X'' with the Unit Interval ''I'' = 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 ightarrow Y, there is a map Sf:SX ightarrow SY defined by Sf( {Link without Title} ):= {Link without Title} . This makes S into a 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 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 .


REDUCED SUSPENSION


If ''X'' is a Pointed Space (with basepoint ''x''0), 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:

:\Sigma X = (X imes I)/(X imes\{0\}\cup X imes\{1\}\cup \{x_0\} imes I).

This is the equivalent to taking ''SX'' and collapsing the line (''x''0 × ''I'') joining the two ends to a single point. The basepoint of Σ''X'' is the Equivalence Class of (''x''0, 0).

One can show that the reduced suspension of ''X'' is Homeomorphic to the Smash Product of ''X'' with the Unit Circle ''S''1.

:\Sigma X \cong S^1 \wedge X

For Well-behaved spaces, such as CW Complex es, 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,

  • \left(\Sigma X,Y ight)\cong \operatorname{Maps}_---\left(X,\Omega Y ight)


  • \left(X,Y ight) stands for continuous maps which preserve basepoints.



SEE ALSO



REFERENCES