One has that the inverse image of a regular value is a manifold by applying the implicit function theorem, see e.g. here.
My question is the following: does the converse hold? I.e., can every manifold (theoretically) be written as the inverse image of a regular value?
I thought I remembered reading this somewhere but now I can't find it. Anyway I was wondering in particular if this result gives an alternative/equivalent definition for manifold which is less intuitive but might be easier to verify in some cases, e.g. spheres.
On the other hand I am not sure how such a weak condition could possibly guarantee that the resulting space is both Hausdorff and second countable, so I am somewhat skeptical that this could actually be true.
To be clear, a reference will suffice for an answer -- this is a dumb question and I don't want to waste too much of anyone's time with it.
Comment: the inverse image of any point under a smooth map $\mathbb{R}^n \to \mathbb{R}^m$ (I assume this is the setting you're working in because you don't want to presuppose a notion of manifold) is a subspace of $\mathbb{R}^n$, so automatically both Hausdorff and second countable. The hard part of the regular value theorem is how to get that the subspace is locally Euclidean, and that's where you need a regular value.
Answer: no. Manifolds that arise in this way must also admit stable framings, which means in particular that their stable characteristic classes must vanish. The simplest example of a manifold not admitting a stable framing is $\mathbb{RP}^2$, because its first Stiefel-Whitney class $w_1$ doesn't vanish; the simplest orientable example is $\mathbb{CP}^2$, because its first Pontryagin class $p_1$ (or, if you prefer, its second Stiefel-Whitney class $w_2$) doesn't vanish.