BundleMap 2002-11-01 17:44:27 2003-08-18 17:48:34 Definition bundle morphism % 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} \usepackage{xypic} % 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 $E_1 \overset{\pi_1}\to B_1$ and $E_2 \overset{\pi_2}\to B_2$ be fiber bundles for which there is a continuous map $f:B_1 \to B_2$ of base spaces. A \emph{bundle map} (or \emph{bundle morphism}) is a commutative square $$ \xymatrix{ E_1 \ar[r]^{\hat{f}} \ar[d]_{\pi_1} & E_2 \ar[d]^{\pi_2} \\ B_1 \ar[r]^{f} & B_2 } $$ such that the induced map $E_1 \to f^{-1}E_2$ is a homeomorphism (here $f^{-1}E_2$ denotes the pullback of $f$ along the bundle projection $\pi_2$).