# 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 |