Talk:Preimage Theorem

As stated, the theorem is nonsense, since a submersion can only have regular values. I suggest the following wording: "Let $\varphi \colon M \to N$ be a smooth map between the smooth manifolds $M,N$ and let $y \in N$ be a regular value of $\varphi$. Then the preimage $\varphi^{-1}(y)$ together with the natural inclusion $\iota:\varphi^{-1}(y) \to M$ is an embedded smooth submanifold of $M$ of dimension $\dim M - \dim N$." --Geometry dude (talk) 00:08, 6 September 2014 (UTC)

This theorem appeared unbacked up by sources some years back in the early days of this site when people were just banging up any old stuff they felt like, before we started structuring it. As such, definitions may not be completely consistent.

Surely though, if a submersion can have only regular values, the theorem is not "nonsense" but tautologous.

Can you find a source for this theorem in the literature? (Not online because the latter may not necessarily be trustworthy.) --prime mover (talk) 00:28, 6 September 2014 (UTC)

It took a while, but I found a source for the theorem in the book by J. Lee "Introduction to smooth manifolds", where it is called "Regular Level Set Theorem". Is it alright then, if I completely rewrite the page including proof and give the source? --Geometry dude (talk) 12:15, 14 September 2014 (UTC)

The preferable course of action in situations like these is that you prepare the page as you envisage it on, say, User:Geometry dude/Sandbox and bring it to our attention (e.g. by posting here again) when you feel like it's finished. We'll then handle migration to the main site. — Lord_Farin (talk) 12:34, 14 September 2014 (UTC)