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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)

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up vote 8 down vote accepted

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.

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