I’m trying to formalize some claims made about simple closed curves in my complex analysis course (its treatment of them is informal since facts like the Jordan curve theorem are so hard to prove.) One fact that is intuitively obvious to me, but which I’m having trouble proving, is the following:
Let $\gamma : [0,1] \to \mathbb{C}$ be a simple closed curve, and let $z_0$ Be a point inside $\gamma$. (I’m assuming we’ve already established the Jordan Curve Theorem.) Then $\gamma$ is homotopic as a closed curve in $\mathbb{C} \setminus \{z_0\}$ to a circle centered at $z_0$, traversed either once counterclockwise or once clockwise.
My initial idea is to continuously project $\gamma$ onto a circle centered at $z_0$, and then show that the projected curve can be “straighted out” on the circle. But I’m not sure how to do that. I’m guessing we need some more sophisticated topological tools — is there a way to do this with the fundamental group? I’d like to avoid using homology if possible, since my knowledge on it is spotty at best. Or if the continuous case is too difficult, I’m willing to settle for the case that $\gamma$ is piecewise $C^1$.