How do I prove this statement about winding numbers and continuous maps?

I have the following problem:

Let $D$ be a disk with boundary circle $C$ and let $f:D\rightarrow \mathbb{R}^2$ be a continuous map. Suppose $P\in \mathbb{R}^2\setminus f(C)$ and the winding number of the restriction $f|_C$ of $f$ to $C$ around $P$ ia not zero. Show that there is a point $Q\in D$ such that $f(Q)=P$

I wanted to prove this by contradiction, i.e. let us assume that $f:D\rightarrow \mathbb{R}^2$ is a continuous map and $C\subset D$ be the boundary circle. Then we can restrict f to $$f|_C:C\rightarrow \mathbb{R}^2$$ which is also continous. Let $P$ be as above s.t. the winding number of $f|_C$ is non zero, i.e. $W(f|_C,P)=W(\gamma,P)\neq 0$ where $$\gamma:[a,b]\rightarrow \mathbb{R}^2$$ is a conitnuous loop and $$\gamma=\phi\circ f|_C$$ with $\phi:[a,b]\rightarrow C$. Now I assume also that forall $Q\in D, f(Q)\neq P$. Since $C\subset D$ we have that also for all $Q\in C, f|_C(Q)\neq P$. Since $\gamma$ is continuous and $[a,b]$ is compact also $\gamma([a,b])$ is compact, but this means that also $f|_C(C)$ is compact.

Now I want to lead this to a contradiction, but I somehow don't see how to procede, could someone help me and show me how I can go further?

Thanks a lot

Solution 1:

The disk $D$ is contractible: fix a point $c \in C \subset D$ and let $F_t(x) : [0, 1] \times D \to D$ be a deformation retraction of $D$ onto $c$ which fixes $c$ at all times $t$. Then the restriction $F_t \rvert_{C}$ of $F_t$ to the boundary circle $C$ is a deformation retraction of $C$ in $D$ onto $c$. That is the same thing as a based homotopy of the loop $F_0 \rvert_{C}$ which ends at the constant loop at $c$.

This means that the composite $f \circ F_0 \rvert_C$ is a based homotopy of $f(C)$ which contracts $f(C)$ to the single point $f(c)$. Finally, we know that the winding number of a loop about a point $P$ is a homotopy invariant of the loop provided that the homotopy does not pass through $P$ (see e.g. here).

So, in the situation of your problem assume that $P \not \in f(D)$. Then the winding numbers $W(f\rvert_C, P)$ and $W([c], P)$ are equal by the explicit homotopy $f \circ F_f \rvert_C$ which we have just constructed (where $[c]$ is the constant loop which stays at $c$). But of course $W([c], P) = 0$, so we conclude that $W(f\rvert_C, P) = 0$ as well. This proves the contrapositive of your claim. (We don't need contradiction here.)

If you're finding the explicit homotopy $F_t$ hard to visualize, by convexity of the disk $D$ we can just spaghetti shlooop up the disk to the point $C$ by (here we think of $D$ as the unit disk in $\mathbb{R}^2$ with $x, c \in \mathbb{R}^2$ vectors of norm $\leq 1$) $$ F_t(x) := (1 - t) x + t c. $$