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'suspension'
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| Title of object: |
suspension |
| Canonical Name: |
Suspension |
| Type: |
Definition |
| Created on: |
2003-02-06 21:56:06.912947-05 |
| Modified on: |
2003-02-06 22:26:32.777291-05 |
| Classification: |
msc:54B99 |
| Defines: |
suspension, reduced suspension, unreduced suspension |
Revision comment (for changes between this and next version):
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Content:
\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 $x, y\in X.$ There is a natural inclusion $X\hookrightarrow \Sigma X$ sending $x$ to the equivalence class $[x,\frac12]$ of $(x,0).$
Given a continuous map $\funcdef{f}{X}{Y},$ there is a map
$\funcdef{Sf}{SX}{SY}$ defined by $\susp f([x,t]):=[f(x),t].$ This makes $S$ into a functor from the category of topological spaces into itself.
An important property of this functor is that it is a left adjoint to the functor $L$ taking a space $X$ to the free loop space $\maps{S^1}{X},$ with the compact-open topology. In other words, $\maps{SX}{Y} \isom \maps{X}{LY}$ naturally.
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} of {\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 is not understood), 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.
The corresponding right adjoint is the functor $\Omega$ assigning to $X$ its (based) loop space $\Omega X = \Omega_{x_0} X.$ In other words, $\bmaps{\susp X}{Y}\isom\bmaps{X}{\Omega Y}$ naturally, where $\bmaps{X}{Y}$ stands for continuous maps which preserve basepoints. |
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