# bundle map

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 bundle map (or 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 $\pi_{2}$).

Title bundle map BundleMap 2013-03-22 13:07:24 2013-03-22 13:07:24 RevBobo (4) RevBobo (4) 5 RevBobo (4) Definition msc 55R10 bundle morphism