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

user45765
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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.

John Hughes
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    Well, perhaps duality akin to the sense that a linear map $T\colon V \to W$ is injective if and only if the dual map $T^\colon W^\to V^*$ is surjective ... – Ted Shifrin Oct 08 '14 at 22:56
  • OK...but what would the dual map be in this case? I still don't see it. (And apparently, the OP isn't convinced by my answer, so he and I both await enlightenment.) – John Hughes Oct 09 '14 at 01:32
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    The transpose of $\begin{bmatrix} 1&0\0&1\0&0\end{bmatrix}$ (which is the linear inclusion of $\Bbb R^2\hookrightarrow\Bbb R^3$) is the projection. – Ted Shifrin Oct 09 '14 at 01:54
  • Sorry...by "in this case" I meant "in the case of submersion/immersion." I knew about the dual for linear maps. :) – John Hughes Apr 17 '16 at 13:44