Suppose that $f:X\rightarrow Y$ is a homotopy equivalence of manifolds. Given a manifold $F$, the pullback construction for $f$ yields a correspondence between isomorphism classes of fibre bundles over $X$ with fibre $F$ and isomorphism classes of fibre bundles over $Y$ with fibre $F$. I am interested in properties of this correspondence. In particular, if $E\rightarrow Y$ is an $F$-bundle over $Y$, then under what conditions is the induced map $f^\*(E)\rightarrow E$ of total spaces a homotopy equivalence? I would appreciate any and all references and suggestions.


Thanks!



Homotopy Equivalences and Induced Correspondences between Fibre Bundles
Well, if your base map $f\colon X \to Y$ is a homotopy equivalence, then the induced map $\tilde{f}\colon f^{\*}E \to E$ will also be a homotopy equivalence.


First, the restriction of $\tilde{f}$ to any fibre is the identity map. Then write down the sequence of homotopy groups for a fibration for both the fibre bundles $f^{\*} E \to X$ and $E \to Y$ together with the homomorphisms induced by $f$ and $\tilde{f}$. Use the 5-lemma to show that $\tilde{f}$ is a weak homotopy equivalence. But because you are dealing with nice spaces (manifolds), a weak homotopy equivalence is an honest homotopy equivalence.


