Let us define the nth smooth homotopy group of a smooth manifold $M$ to be the group $\pi_n^\infty(S^k)$ of smooth maps $S^n \to S^k$ modulo smooth homotopy. Of course, some care must be taken to define the product, but I don't think this is a serious issue. The key is to construct a smooth map $S^n \to S^n \lor S^n$ (regarded as subspaces of $\mathbb{R}^{n+1}$) which collapses the equator to a point; we then define the product of two (pointed) maps $f, g: S^n \to S^k$ to be the map $S^n \lor S^n \to S^k$ which restricts to $f$ on the left half and $g$ on the right half. To accomplish this, use bump functions to bend $S^n$ into a smooth "dumbell" shape consisting of a cylinder $S^{n-1} \times [0,1]$ with two large orbs attached to the ends, and retract $S^{n-1}$ to a point while preserving smoothness at the ends. Then retract $[0,1]$ to a point, and we're done.

Question: is the natural "forget smoothness" homomorphism $\phi: \pi_n^\infty(S^k) \to \pi_n(S^k)$ an isomorphism? If not, what is known about $\pi_n^\infty(S^k)$ and what tools are used?

In chapter 6 of "From Calculus to Cohomology", Madsen and Tornehave prove that every continuous map between open subsets of Euclidean spaces is homotopic to a smooth map. Thus every continuous map $f$ between smooth manifolds is "locally smooth up to homotopy", meaning that every point in the source has a neighborhood $U$ such that $f|_U$ is homotopic to a smooth map. However it is not clear to me that the local homotopies can be chosen in such a way that they glue together to form a global homotopy between $f$ and a smooth map. This suggests that $\phi$ need not be surjective.

In the same reference as above, it is shown that given any two smooth maps between open subsets of Euclidean spaces which are continuously homotopic, there is a smooth homotopy between them. As above this says that two smooth, continuously homotopic maps between smooth manifolds are locally smoothly homotopic, but I again see no reason why the local smooth homotopies should necessarily glue to form a global smooth homotopy. This suggests that $\phi$ need not be injective.

I am certainly no expert on homotopy theory, but I have read enough to be surprised that this sort of question doesn't seem to be commonly addressed in the basic literature. This leads me to worry that my question is either fatally flawed, trivial, useless, or hopeless. Still, I'm retaining some hope that something interesting can be said.