suspensionSuspension2003-02-06 21:56:062006-02-15 03:45:16Definitionsuspensionreduced suspensionbased suspensionunreduced suspensionunbased suspension% used for TeXing text within eps files
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\newcommand{\susp}{\Sigma}\section{The unreduced suspension}
Given a topological space $X,$ the {\em suspension} of $X,$ often denoted by $SX,$ is defined to be the quotient space $X\cross[0,1]/\sim,$ where $(x,0)\sim(y,0)$ and $(x,1)\sim(y,1)$ for any $x, y\in X.$
Given a continuous map $\funcdef{f}{X}{Y},$ there is a map
$\funcdef{Sf}{SX}{SY}$ defined by $Sf([x,t]):=[f(x),t].$ This makes $S$ into a functor from the category of topological spaces into itself.
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 {\em unreduced}, {\em unbased} or {\em free} suspension of $X,$ to distinguish it from the reduced suspension described below.
\section{The reduced suspension}
If $(X,x_0)$ is a based topological space, the {\em reduced suspension} of $X,$ often denoted $\susp X$ (or $\susp_{x_0} X$ when the basepoint needs to be explicit), is defined to be the quotient space $X\times[0,1]/(X\cross\set{0}\cup X\cross\set{1}\cup\set{x_0}\cross[0,1].$ Setting the basepoint of $\susp X$ to be the equivalence class of $(x_0,0),$ the reduced suspension is a functor from the category of based topological spaces into 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, $\bmaps{\susp X}{Y}\isom\bmaps{X}{\Omega Y}$ naturally, where $\bmaps{X}{Y}$ stands for continuous maps which preserve basepoints.
The reduced suspension is also known as the {\em based\/} suspension.