# suspension isomorphism

###### Proposition 1.

Let $X$ be a topological space^{}. There is a natural isomorphism

$$s:{H}_{n+1}(SX)\to {H}_{n}(X),$$ |

where $S\mathit{}X$ stands for the unreduced suspension of $X\mathrm{.}$

If $X$ has a basepoint, there is a natural isomorphism

$$s:{\stackrel{~}{H}}_{n+1}(\mathrm{\Sigma}X)\to {\stackrel{~}{H}}_{n}(X),$$ |

where $\mathrm{\Sigma}\mathit{}X$ is the reduced suspension.

A similar proposition^{} holds with homology^{} replaced by cohomology.

In fact, these propositions follow from the Eilenberg-Steenrod axioms without the dimension axiom, so they hold for any generalized (co)homology theory in place of integral (co)homology.

Title | suspension isomorphism |
---|---|

Canonical name | SuspensionIsomorphism |

Date of creation | 2013-03-22 13:25:58 |

Last modified on | 2013-03-22 13:25:58 |

Owner | antonio (1116) |

Last modified by | antonio (1116) |

Numerical id | 4 |

Author | antonio (1116) |

Entry type | Theorem |

Classification | msc 55N99 |

Related topic | Suspension |