homotopy invariance HomotopyInvariance 2004-06-15 16:06:17 2004-06-15 16:06:17 Definition homotopy invariant % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $\cal F$ be a functor from the category of topological spaces to some category $\cal C$. Then $\cal F$ is called {\em homotopy invariant} if for any two homotopic maps $f,g\colon X\to Y$ between topological spaces $X$ and $Y$ the morphisms ${\cal F}f$ and ${\cal F}g$ in $\cal C$ induced by $\cal F$ are identical. Suppose $\cal F$ is a homotopy invariant functor, and $X$ and $Y$ are homotopy equivalent topological spaces. Then there are continuous maps $f\colon X\to Y$ and $g\colon Y\to X$ such that $g\circ f\simeq{\rm id}_X$ and $f\circ g\simeq{\rm id}_Y$ (i.e. $g\circ f$ and $f\circ g$ are homotopic to the identity maps on $X$ and $Y$, respectively). Assume that $\cal F$ is a covariant functor. Then the homotopy invariance of $\cal F$ implies $$ {\cal F}g\circ{\cal F}f={\cal F}(g\circ f)={\rm id}_{{\cal F}X} $$ and $$ {\cal F}f\circ{\cal F}g={\cal F}(f\circ g)={\rm id}_{{\cal F}Y}. $$ From this we see that ${\cal F}X$ and ${\cal F}Y$ are isomorphic in $\cal C$. (The same argument clearly holds if $\cal F$ is contravariant instead of covariant.) An important example of a homotopy invariant functor is the fundamental group $\pi_1$; here $\cal C$ is the category of groups.