# suspension

## 1 The unreduced suspension

Given a topological space $X,$ the of $X,$ often denoted by $SX,$ is defined to be the quotient space $X\times[0,1]/\sim,$ where $(x,0)\sim(y,0)$ and $(x,1)\sim(y,1)$ for any $x,y\in X.$

Given a continuous map $f:X\rightarrow Y,$ there is a map $Sf:SX\rightarrow SY$ 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 unreduced, unbased or free suspension of $X,$ to distinguish it from the reduced suspension described below.

## 2 The reduced suspension

If $(X,x_{0})$ is a based topological space, the reduced suspension of $X,$ often denoted $\Sigma X$ (or $\Sigma_{x_{0}}X$ when the basepoint needs to be explicit), is defined to be the quotient space $X\times[0,1]/(X\times\left\{0\right\}\cup X\times\left\{1\right\}\cup\left\{x_% {0}\right\}\times[0,1].$ Setting the basepoint of $\Sigma X$ 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, $\mathop{\mathrm{Maps}_{*}}\left(\Sigma X,Y\right)\cong\mathop{\mathrm{Maps}_{*% }}\left(X,\Omega Y\right)$ naturally, where $\mathop{\mathrm{Maps}_{*}}\left(X,Y\right)$ 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