Homotopy invariance of winding number

A homotopy through closed paths avoiding the base point cannot change the integer winding number.

If closed continuous paths $\gamma ,\varphi$ satisfy $|\gamma (t)|>|\varphi (t)|$ for every $t$, then $H(s,t)=\gamma (t)+s\varphi (t)$ never vanishes. It is therefore a homotopy through closed paths in $\mathbb{C}\setminus \{0\}$, and

For a polynomial $P(z)=a_n{z}^n+\cdots +a_0$ of positive degree, on a sufficiently large circle its leading term dominates the remaining terms. The dominated-perturbation lemma therefore gives winding number $n$ to the loop $P(R{e}^{2\pi it})$. If $P$ had no zero, radial contraction of the input circle would map under $P$ to a homotopy with a constant loop in $\mathbb{C}\setminus \{0\}$, which has winding number zero. This contradiction proves the Fundamental theorem of algebra.

If a continuous retraction from a closed disc to its boundary existed, applying it to a contraction of the boundary circle inside the disc would give a homotopy in $\mathbb{C}\setminus \{0\}$ from a loop of winding number one to a constant loop of winding number zero. Homotopy invariance of winding number rules this out.