Every map $\alpha: \partial I^n \to \mathbb{R}^m\setminus\{0\}$ could be extended to a map $\tilde{\alpha}: I^n \to \mathbb{R}^m\setminus\{0\}$

Question

Are the following two propositions true?

Every map $\alpha: \partial I^n \to \mathbb{R}^m\setminus\{0\}$ could be extended to a map $\tilde{\alpha}: I^n \to \mathbb{R}^m\setminus\{0\}$ such that $\tilde\alpha|_{\partial I^n} = \alpha$

Every map $\alpha: \partial I^n \to \mathbb{R}^m$ could be extended to a map $\tilde{\alpha}: I^n \to \mathbb{R}^m$ such that $\tilde\alpha|_{\partial I^n} = \alpha$ and $\alpha(\mathring {I^n}) \subseteq \mathbb{R}^m\setminus\{0\}$

In both cases I assume $n < m$.

For $n = 1$ the first proposition only asserts that the space is arcwise-connected and that's true for $m > 1$. The second proposition is also true for $n = 1$. I think that the first proposition, for every $n < m$, is true because $H_{n-1}(\mathbb{R}^m\setminus\{0\}) \cong 0$.

Answer 1

The first is (basically) just a question about the homotopy groups of $\mathbb{R}^m - \{0\}$. Your question is equivalent to asking if $\pi_{n-1}(S^{m-1})=0$ for $n<m$. This is true, but not particularly easy. Hatcher proves it in his section on the homotopy groups.

The second question is trickier. It is equivalent to asking if there is a homotopy from the initial map of your sphere to a constant one, where at no time after $0$ do the slice maps map your sphere into the origin. I'm sure that if the map was smooth one could give an argument in the affirmative because of the dimension condition given. However, I am not so sure in the topological case because of the existence of space filling curves.