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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!

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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.

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This should be sufficient since each of my spaces has the homotopy type of a CW-complex. Whitehead's Theorem should then give the desired result. I was thinking there might be a reasonably elementary proof that did not invoke Whitehead's Theorem. Perhaps this is not true for spaces not homotopy-equivalent to CW-complexes. Do you know of a counter-example? –  Peter Crooks Feb 28 '13 at 16:08
    
Yes, by a nice space I meant one with the homotopy type of a CW-complex :-) I suspect you can prove the homotopy equivalence of the total space map without going through the weak homotopy equivalence. Probably just by looking at what is happening when the base map is being homotoped, but I have to admit I didn't think it through enough. –  Oldřich Spáčil Feb 28 '13 at 16:28
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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.

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With regard to Spanier's book p. 95 (first edition) he defines a few lines above 12 Theorem an "extended lifting function" $\Lambda". I suggested a to him another formula to make sure the functionwas well defined, and this appeared in the second edition. But I confess I gave up trying to prove this modified function was continuous. Has anyone written out a proof? I also refer people to the paper by Dyer and Eilenberg, Globalizing fibrations by schedules. Fund. Math. 130 (1988), 125--136. The right pro-perness for Hurewicz fibrations was first observed, I think, in R. Brown and P.R. –  Ronnie Brown Mar 1 '13 at 14:35
    
(ran out of space, so continue) ``Coglueing homotopy equivalences'', Math. Z. 113 (1970) 313-362. This was taken as the dual of the "glueing theorem" in the 1968 edition of my book now available as "Topology and Groupoids". These proofs have the advantage of giving control over the homotopies involved. –  Ronnie Brown Mar 1 '13 at 14:40
    
This is fantastic advice! I really appreciate the input from each of you! –  Peter Crooks Mar 1 '13 at 16:18
    
@Ronnie: Thank you very much for your comment. I had actually heard that before about the Hurewicz theorem in Spanier's book, but had forgotten about it. I must admit I have never checked that section of Spanier's book --- along with many other sections :). Also, I was not aware of the original references for the right properness of the Strom model structure. Thanks! –  Ricardo Andrade Mar 1 '13 at 22:26
    
@PDC: It was my pleasure. –  Ricardo Andrade Mar 1 '13 at 22:45
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