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In Hatcher's Algebraic Topology book it is noted after 4.61 that:

fiber preserving map + homotopy equivalence $\Rightarrow$ fiber homotopy equivalence.

Question:

Could there be two fibrations over the same base space where the total spaces are homotopy equivalent, but there is no fiber homotopy equivalence between them? (and therefore also no fiber preserving map)

If so, I would be glad to have a simple example.

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4 Answers 4

up vote 7 down vote accepted

You can fiber a circle over a circle in many ways.

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Could you please give some more detail? –  Shlomi A Aug 11 '12 at 18:01
    
A two to one map (covering space) from circle to circle is a fibration, in which every fiber has two points. The identity map is another, in which every fiber has one point. –  Tom Goodwillie Aug 11 '12 at 19:03
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Let X be any non-contractible space. Let the base of your fibration be the disjoint union of countably many copies of X and countably many copies of the point. Let one fibration be the identity and the other be the identity over every component but one point; put X above that point. The two spaces are both abstractly homeomorphic to the base, but a fiber homotopy equivalence would have to be a homotopy equivalence between X and the point over that particular component.

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Right. Thanks! :) –  Shlomi A Aug 11 '12 at 18:05
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There are lots of such examples, but here is the simplest one I know. Let $M_{p,q}$ be the total space of the principal circle bundle over $S^2\times S^2$ with Euler class $(p, q)$.

If $p,q$ are relatively prime, then it is known that $M_{p,q}$ is diffeomorphic to $S^2\times S^3$. Namely, Smale proved in his paper "On the structure of 5-manifolds" that the diffeomorphism type of closed, simply-connected, spin 5-manifolds is determined by the second cohomology which is $\mathbb Z$ for $M_{p,q}$ and also for $S^2\times S^3$. (If you have trouble showing $M_{p,q}$ satisfies the above conditions, see Wang-Ziller's paper "Einstein metrics on principal torus bundles."

On the other hand, the fiber homotopy equivalence in this case preserves the Euler class (up to sign).

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I think James-Whitehead are quite relevant:

MR0068836 Reviewed James, I. M.; Whitehead, J. H. C. The homotopy theory of sphere bundles over spheres. II. Proc. London Math. Soc. (3) 5, (1955). 148–166.

MR0061838 Reviewed James, I. M.; Whitehead, J. H. C. The homotopy theory of sphere bundles over spheres. I. Proc. London Math. Soc. (3) 4, (1954). 196–218.

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