A homotopy equivalence between total spaces in a (Hurewicz) fibration which is not a fiber homotopy equivalence 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.
 A: 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.
A: You can fiber a circle over a circle in many ways.
A: 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).
A: 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. 
