1 The unreduced suspension

Given a topological spaceMathworldPlanetmath X, the suspensionMathworldPlanetmath of X, often denoted by SX, is defined to be the quotient spaceMathworldPlanetmath X×[0,1]/, where (x,0)(y,0) and (x,1)(y,1) for any x,yX.

Given a continuous map f:XY, there is a map Sf:SXSY defined by Sf([x,t]):=[f(x),t]. This makes S into a functorMathworldPlanetmath from the category of topological spaces into itself.

Note that SX is homeomorphicMathworldPlanetmath to the join XS0, where S0 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.

2 The reduced suspension

If (X,x0) is a based topological spacePlanetmathPlanetmath, the reduced suspension of X, often denoted ΣX (or Σx0X when the basepoint needs to be explicit), is defined to be the quotient space X×[0,1]/(X×{0}X×{1}{x0}×[0,1]. Setting the basepoint of ΣX to be the equivalence classMathworldPlanetmathPlanetmath of (x0,0), the reduced suspension is a functor from the categoryMathworldPlanetmath of based topological spaces into itself.

An important property of this functor is that it is a left adjoint to the functor Ω taking a (based) space X to its loop spaceMathworldPlanetmath ΩX. In other words, Maps*(ΣX,Y)Maps*(X,ΩY) naturally, where Maps*(X,Y) stands for continuous maps which preserve basepoints.

The reduced suspension is also known as the based suspension.

Title suspension
Canonical name Suspension
Date of creation 2013-03-22 13:25:37
Last modified on 2013-03-22 13:25:37
Owner antonio (1116)
Last modified by antonio (1116)
Numerical id 10
Author antonio (1116)
Entry type Definition
Classification msc 54B99
Related topic Cone
Related topic LoopSpace
Related topic Join3
Related topic SuspensionIsomorphism
Defines suspension
Defines reduced suspension
Defines based suspension
Defines unreduced suspension
Defines unbased suspension