In the category of smooth manifolds (without corners), what are some examples of Hurewicz fibrations which are not fiber bundles?


The minimal topological example I know is to project the standard 2-simplex onto the $x$-axis. The right-most fiber degenerates into a point while the others are homeomorphic to a closed interval.


I don't understand how to produce such a "dimensional degeneration" phenomenon in the smooth world. In fact this seems sort of impossible to me: a Hurewicz fibration is a submersion and the vertical bundle of a submersion has locally constant rank, so the fibers are homotopy equivalent equidimensional embedded submanifolds which foliate the source (assume the base is connected).


I don't see what other kind of degeneration (other than dimensional) might preclude a fibration from being a fiber bundle.


Out of helplessness, since fibrations are submersions, I was tempted to examine submersions which are not fiber bundles. The classical example $\mathbb R^2\to \mathbb R,\; (x,y)\mapsto(x^2-1)e^y$ is not a Hurewicz fibration because fibers of negative numbers are connected while those of non-negative numbers are disconnected. I don't know any other examples.



Examples of smooth Hurewicz fibrations which are not smooth fiber bundles
Some poking around led to an example in G. Meigniez, [*Submersions, fibrations, and bundles*](https://www.ams.org/journals/tran/2002-354-09/S0002-9947-02-02972-0/S0002-9947-02-02972-0.pdf), Trans. Amer. Math. Soc. 354 (2002), 3771-3787. It's Example 21 in that paper and described briefly as follows: Let $W\subset\mathbb R^3$ be the Whitehead manifold, an open, contractible subset not homeomorphic to $\mathbb R^3$. Let $E$ be the set of $(x,y,z,t)\in\mathbb R^4$ with $(x,y,z)\in W$ or $t \neq 0$. Then the projection $\pi\colon E\to \mathbb R$, $\pi(x,y,z,t) = t$, is a smooth submersion, a fibration (with contractible fibers), but not locally trivial because the fiber over $0$ is not homeomorphic to the others.


That paper also refers to other counterexamples given in S. Ferry, [*Alexander duality and Hurewicz fibrations*](https://www.ams.org/journals/tran/1991-327-01/S0002-9947-1991-1028308-9/S0002-9947-1991-1028308-9.pdf), Trans. Amer. Math Soc. 327 (1991), 201-219.


