Homotopy Equivalences and Induced Correspondences between Fibre Bundles

Question

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!

Answer 1

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.

Answer 2

Here is a quick argument which proves homotopy equivalence directly. First, the pullback of a homotopy equivalence along a Hurewicz fibration is again a homotopy equivalence. Further, by the well-known uniformization theorem of Hurewicz, a locally trivial fibre bundle $E\to Y$ over a paracompact Hausdorff space $Y$ (such as a manifold) is a Hurewicz fibration. We thus conclude that if $f:X\to Y$ is a homotopy equivalence then the map $f^\ast E\to E$ is again a homotopy equivalence.

A few references and remarks

There are many references for the uniformization theorem of Hurewicz, both in articles and in books. For example, in article form, there is the original article by Hurewicz, this article by Dold, and this other article in a slightly more modern language. In book form, there is Spanier's book Algebraic topology, the book A concise course in algebraic topology by May, and tom Dieck's Algebraic topology.

The fact that the pullback of a homotopy equivalence along a Hurewicz fibration is again a homotopy equivalence is called — in the jargon of model categories — the right properness of the Strøm model structure on the category of topological spaces. Actually, any model category in which all objects are fibrant is right proper, and that includes the Strøm model structure. Nevertheless, a direct, elementary proof that the Strøm model structure is right proper can be given using solely the covering homotopy property for Hurewicz fibrations.