smooth homotopy on exotic R^4

Question

Take an exotic $\mathbb{R}^4$ i.e. $V = (\mathbb{R}^4,d)$ such that $V$ is not diffeomorphic to $\mathbb{R}^4$ with standard metric.

Is it true (obvious?) that any two smooth maps $f_1, f_2: S^k \to V$ are equivalent via smooth homotopy?

edited: (for any k)

Answer 1

Yes, since smooth maps which are (continuously) homotopic are always smoothly homotopic. See Kosinski's "Differential Manifolds", Theorem III.2.5 and Corollary III.2.6.