I am not an expert in Differential Topology, so let me apologize if this question admits a straightforward answer. I checked some standard references, but I could not find one.


Let $M$ be a smooth $n$-manifold with boundary $N:= \partial M$, and assume that there exists a *continuous* retraction of $M$ onto $N$, namely, a continuous map $r \colon M \to N$ that is the identity on $N$.



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> **Question.** Does there exist a *smooth* retraction $s \colon M \to N$? And what about a smooth retraction homotopic to $r$?
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I am aware that, by Whitney Approximation Theorem, $r \colon M \to N$ is homotopic to a smooth map, but I do not see how to conclude from this that it is homotopic to a smooth retraction.


**Motivation.** This question arose when I was trying to mimic the standard proof of Brouwer Fixed Point Theorem (for *continuous* maps $f \colon \mathbb{D}^n \to \mathbb{D}^n$) by using de Rham cohomology, instead of singular homology. In the homology setting, the starting point is the non-existence of a continuous retraction $r \colon \mathbb{D}^n \to S^{n-1}$, which is obtained by contradiction looking at the functorial group homomorphisms induced by $r$ and by the inclusion $i \colon S^{n-1} \to \mathbb{D}^n$. But de Rham cohomology is only functorial with respect to *smooth* maps...



Is every retraction homotopic to a smooth retraction?
Using a collar of the boundary we resort to the case when $r\in C^0(M, \partial M)$ is given by the projection $\partial M\times I \to \partial M$ over the collar .


Since $r$ is smooth on an open neighbourhood of $\partial M$, for any $\epsilon>0$ we can find a smooth $g\in C^\infty(M,\partial M)$ such that $g=r$ in a neighbourhood of $\partial M$ and $|g-r|<\epsilon$.


For a reference see Prop. 3.11 of Milnor's lectures on Differential Topology 1958 (notes taken by James Munkres).


