Let $E \to M$ be a fiber bundle and $f,g \colon N \to M$.
- We know (e.g. from Steerod) that if $f$ and $g$ are homotopic, then the pullback bundles are equivalent bundles over $N$, $f^\ast E \cong g^\ast E$.
- What about the reverse implication? If we assume that $f^\ast E \cong g^\ast E$, in what cases does this imply that $f$ and $g$ are homotopic?