# Is $π:\mathcal{C}^∞(M,N)→\mathcal{C}^∞(S,N)$, $π(f)=f|_S$ a quocient map in the $\mathcal{C}^1$ topology?

This question was previously posted on MSE.

Let $$M, N$$ be smooth connected manifolds (without boundary), where $$M$$ is a compact manifold, so we can put a topology in the space $$\mathcal C^\infty(M, N)$$ using $$\mathcal{C}^1$$ Whitney Topology.

Now, consider $$S\subset M$$ a compact submanifold of $$M$$ with boundary such that $$\text{dim}S=\text{dim}M$$, using the same process we can put a topology in $$\mathcal C^\infty(S,N)$$ using the $$\mathcal{C}^1$$ Whitney Topology. There is a natural continous projection of $$\mathcal C^\infty(M, N)$$ on $$\mathcal C^\infty(S,N)$$, definided by

\begin{align*} \pi: \mathcal C^\infty(M, N) &\to \mathcal C^\infty(S,N)\ f&\mapsto \left.f\right|_{S}. \end{align*}

My Question: Is $$\pi$$ an open map or at least a quocient map?

$$\mathcal{C}^1$$-Whitney Topology is also called $$\mathcal{C}^1$$-strong topology.

As noticed for the user Adam Chalumeau, on the book “Morris W. Hirsh Differential Topology” there is the following exercise

[Exercise 16, page 41]: Let $$M, N$$ be $$\mathcal{C}^r$$ manifolds. Let $$V⊂M$$ be an open set then

• The restriction map $$δ:\mathcal{C}^r(M,N)→\mathcal{C}^r(V,N)$$ $$δ(f)=f|V$$ is continuous for the weak topology, but not always for the strong.

• $$δ$$ is open for the strong topologies, but not always for the weak”.

Since our $$M$$ is compact weak topology = strong topology. However, I don’t know how to solve this exercise let alone adapt such proof to the case that I want.