suspension

1 The unreduced suspension

Given a topological space $X,$ the suspension 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 $f:X\to Y,$ there is a map $Sf:SX\to SY$ 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 $\mathrm{\Sigma }X$ (or ${\mathrm{\Sigma }}_{x_0}X$ 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 $\mathrm{\Sigma }X$ 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 $\mathrm{\Omega }$ taking a (based) space $X$ to its loop space $\mathrm{\Omega }X$. In other words, ${\mathrm{Maps}}_{*}(\mathrm{\Sigma }X,Y)\cong {\mathrm{Maps}}_{*}(X,\mathrm{\Omega }Y)$ naturally, where ${\mathrm{Maps}}_{*}(X,Y)$ 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