Submersion and immersion

Question

I googled wiki about submersion and immersion. Wiki states that submersion is dual to immersion. I wonder where this duality relationship comes from.

Answer 1

I think it's a pretty sloppy notion of duality: If you put a $k$-manifold in $R^n$ with maximal derivative rank, there are three choices:

$$ k < n \\ k = n \\ k > n $$ The middle choice is generally uninteresting (at least for compact manifolds without boundary). The other two correspond to "immersion" and "submersion".

By "sloppy", I mean that there's no "dualizing" map described -- there's no way to take a submersion and generate a corresponding immersion, which if it is dualized brings you back to the original submersion; not only does the article not mention such a map, I don't know of one, either (I'd like to!). Contrast that with, say, Poincare duality, where the dualizing map is quite clear.