Proving Brouwer's fixed point theorem using fundamental groups

Question

I am writing my bachelor thesis on the fundamental group $\pi_1(X)$ and homotopy theory.
Now I was wondering if it is possible to prove Brouwer's fixed point Theorem in arbitrary dimensions using only $\pi_1$ and no higher homotopy groups.
Since $\pi_1(S^n) = 0$ for $n \geq 2$ I don't seem to get a contradiction to having a retract $S^{n -1} \to D^n$ for $n \geq 3$.

Answer 1

I think you answered your question yourself. From the perspective of $\pi_1$ the spaces $\Bbb S^n$ and $\Bbb D^n$ with $n\geq 2$ are the same, so you cannot distinguish them with it. In particular you cannot construct a contradiction to the existence of a retraction.