If $p<n$, is this possible? I am confused about this. I am trying to prove that the i-th singular homotopy group of the n-sphere is a subset of the i-th homotopy group of $\mathbb{R}^n$ but I am having trouble as using partition functions to "glue" the coordinate charts applied to cycles doesn't seem to guarantee an injective map. Any hints on this would be greatly appreciated. The goal is to use this to show that if $i<n$, the i-th homotopy group of the sphere is just 0.
EDIT: I forgot to mention that the map has to be $C^\infty$! Sorry!
Yes there exists such a surjective map!
Actually for any $n\gt 0$ there exists a surjective continuous map from the $1$-simplex, aka the unit interval $[0,1]$, to $\mathbb R^n$ or $S^n$: these are examples of the Peano phenomenon of space-filling curves.
The most general result in that area is the Hahn-Mazurkiewicz theorem:
Every second-countable, compact, connected, locally connected space is a continuous image of the unit interval .
To prove that $\pi_i(S^n)=0$ for $0\lt i\lt n$ (which is indeed true!) , you have to approximate maps $S^i\to S^n$ by smooth maps.