I hope this question isn't too open-ended for MO --- it's not my favorite type of question, but I do think there could be a good answer. I will happily CW the question if commenters want, but I also want answerers to pick up points for good answers, so...


Let $X,Y$ be smooth manifolds. A smooth map $f: Y \to X$ is a **bundle** if there exists a smooth manifold $F$ and a covering $U\_i$ of $X$ such that for each $U\_i$, there is a diffeomorphism $\phi\_i : F\times U\_i \overset\sim\to f^{-1}(U\_i)$ that intertwines the projections to $U\_i$. This isn't my favorite type of definition, because it demands existence of structure without any uniqueness, but I don't want to define $F,U\_i,\phi\_i$ as part of the data of the bundle, as then I'd have the wrong notion of morphism of bundles.


A definition I'm much happier with is of a **submersion** $f: Y \to X$, which is a smooth map such that for each $y\in Y$, the differential map ${\rm d}f|\_y : {\rm T}\_y Y \to {\rm T}\_{f(y)}X$ is surjective. I'm under the impression that submersions have all sorts of nice properties. For example, preimages of points are embedded submanifolds (maybe preimages of embedded submanifolds are embedded submanifolds?).


So, I know various ways that submersions are nice. Any bundle is in particular a submersion, and the converse is true for proper submersions (a map is **proper** if the preimage of any compact set is compact), but of course in general there are many submersions that are not bundles (take any open subset of $\mathbb R^n$, for example, and project to a coordinate $\mathbb R^m$ with $m\leq n$). But in the work I've done, I haven't ever really needed more from a bundle than that it be a submersion. Then again, I tend to do very local things, thinking about formal neighborhoods of points and the like.


So, I'm wondering for some applications where I really need to use a bundle --- where some important fact is not true for general submersions (or, surjective submersions with connected fibers, say).



Why should I prefer bundles to (surjective) submersions?
One would be that a fibre bundle $F \to E \to B$ has a homotopy long exact sequence


$$ \cdots \to \pi\_{n+1} B \to \pi\_n F \to \pi\_n E \to \pi\_n B \to \pi\_{n-1} F \to \cdots $$


This isn't true for a submersion, for one, the fibre in a submersion does not have a consistent homotopy-type as you vary the point in the base space.


