Is there an elementary proof of the triviality of the first homotopy groups of spheres (i.e. the statement that for $1\leq k<n,\;\pi_k(S^n)=0$)? By elementary I mean without using the tool of cellular approximation or Hurewicz theorem.
(This question is a follow up to the recent question asking for an elementary proof in the case $k=1$ (which is needed to apply the Hurewicz theorem anyways), so I'm only interested in the case $k>1$)
Given a continuous function $f:S^k \to S^n$, find a continuously differentiable function $g:S^k \to S^n$ so that $|g(x) - f(x)| \le 1/10$. Show that $f$ and $g$ are homotopic. Also show that $g$ cannot be onto (i.e. there are no space filling continuously differentiable maps). Use a point in $S^n \setminus g(S^k)$ to create the homotopy to the constant map.