# homotopy invariance

Let $\mathcal{F}$ be a functor^{} from the category of topological spaces to some category^{} $\mathcal{C}$. Then $\mathcal{F}$ is called homotopy invariant if for any two homotopic maps $f,g:X\to Y$ between topological spaces^{} $X$ and $Y$ the morphisms^{} $\mathcal{F}f$ and $\mathcal{F}g$ in $\mathcal{C}$ induced by $\mathcal{F}$ are identical.

Suppose $\mathcal{F}$ is a homotopy invariant functor, and $X$ and $Y$ are homotopy equivalent topological spaces. Then there are continuous maps $f:X\to Y$ and $g:Y\to X$ such that $g\circ f\simeq {\mathrm{id}}_{X}$ and $f\circ g\simeq {\mathrm{id}}_{Y}$ (i.e. $g\circ f$ and $f\circ g$ are homotopic^{} to the identity maps on $X$ and $Y$, respectively). Assume that $\mathcal{F}$ is a covariant functor. Then the homotopy invariance of $\mathcal{F}$ implies

$$\mathcal{F}g\circ \mathcal{F}f=\mathcal{F}(g\circ f)={\mathrm{id}}_{\mathcal{F}X}$$ |

and

$$\mathcal{F}f\circ \mathcal{F}g=\mathcal{F}(f\circ g)={\mathrm{id}}_{\mathcal{F}Y}.$$ |

From this we see that $\mathcal{F}X$ and $\mathcal{F}Y$ are isomorphic^{} in $\mathcal{C}$. (The same argument clearly holds if $\mathcal{F}$ is contravariant instead of covariant.)

An important example of a homotopy invariant functor is the fundamental group^{} ${\pi}_{1}$; here $\mathcal{C}$ is the category of groups.

Title | homotopy invariance |
---|---|

Canonical name | HomotopyInvariance |

Date of creation | 2013-03-22 14:24:51 |

Last modified on | 2013-03-22 14:24:51 |

Owner | pbruin (1001) |

Last modified by | pbruin (1001) |

Numerical id | 4 |

Author | pbruin (1001) |

Entry type | Definition |

Classification | msc 55Pxx |

Related topic | HomotopyEquivalence |

Defines | homotopy invariant |