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Let $f:N\rightarrow M$ be a surjective submersion.

Let $g:W\rightarrow M$ be a smooth map.

Let $w\in W$. We have $g(w)\in M$. As $f$ is surjective, there exists $n\in N$ such that $f(n)=g(w)$.

Define $p:W\rightarrow N$ such that $p(w)=n$ where $n\in N$ is such that $f(n)=g(w)$.

I am wondering if this map $p$ is smooth.

Any hints are welcome.

I am seeing the situation as following case

enter image description here

As $f$ is a submersion, pull back $W\times_MN$ is a smooth manifold.

As $f$ is surjective, given $w\in W$ there exists $n\in N$ : $f(n)=g(w)$, giving an element $(w,n)\in W\times _MN$ which gives a obvious map $W\rightarrow W\times_M N$ and it is clear that $pr_2$ composed with this map is equal to $p$.

As $pr_2$ map is smooth and as $W\rightarrow W\times_MN$ is smooth (being an inclusion) the composition $p:W\rightarrow N$ is smooth.

Flaw in my idea is that the map $W\rightarrow W\times_MN$ is not an inclusion. It is not of the form $w\mapsto (w,n)$ for some fixed $n$. It is of the form $w\mapsto (w,n_w)$ which is not inclusion.

Any suggestions to make this work are welcome.

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The map $p$ is not well defined, because typically there are many choices of $n$ such that $f(n) = g(w)$. As $f$ surjective, there always exists such a map by the axiom of choice, but there's no reason to expect that it should be smooth or even continuous. And in fact, there are many situations in which there is no such map that is continuous.

Consider this example: Let $N=\mathbb R$, $M=W=\mathbb S^1$ (thought of as a submanifold of $\mathbb C$), and define $f\colon \mathbb R\to \mathbb S^1$ and $g\colon \mathbb S^1\to \mathbb S^1$ by \begin{align*} f(x) &= e^{ix},\\ g(z) &= z. \end{align*} There is no continuous map $p\colon \mathbb S^1\to \mathbb R$ such that $e^{ip(z)} = z$ for all $z\in \mathbb S^1$.

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  • $\begingroup$ thanks for the answer.. Though my first trial turn out to be misleading, I still had some hope.. Can you tell me what do you think might be the obvious thing that I was missing... I could not yet figure out why my claim could be false... I want to know what could be missing with out going for an example... If this is too much to ask, I am ok with accepting this much for now.. It would be good if you can say something.. $\endgroup$ – user537667 Sep 17 '18 at 17:44
  • $\begingroup$ @PraphullaKoushik: I'm not sure what you're asking now. You've already found a flaw in your argument, and my counterexample shows that your claim is false. What more do you want? $\endgroup$ – Jack Lee Sep 17 '18 at 18:41
  • $\begingroup$ I am sorry for the confusion... I am asking that, before constructing example what made you to think claim is false? Is it better now? $\endgroup$ – user537667 Sep 18 '18 at 4:09

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