I'm new here and to the differential geometry. I find a problem when I'm trying to prove Brouwer's fixed point theorem using topology.
Suppose $f: D^{n}\to D^{n}$ has no fixed point, define $F: D^{n}\to S^{n-1}$ such that $F(x)$ denotes the intersection of $S^{n-1}$ and a line from $f(x)$ through $x$. In addition, define $g: S^{n-1}\to D^{n}$ as an inclusion map.
I think we can construct a map $H(t,x)\equiv t\cdot id_{D^n}+(1-t)g\circ F$, indicating the $S^{n-1}$ is a deformation retract of $D^n$ and thus they should have the same homotopy type. We can reach the contradiction that $\mathbb{Z}=\pi_{n-1}(S^{n-1})=\pi_{n-1}(D^n)=0$.
I think it might be correct in 2-dimensional case. Why do we have to use homology groups in higher dimension? Did I neglect some subtleties?
Thanks so much!