A few questions regarding the winding number.

Question

A few questions baout the winding numbers:

1. Why do two homotopic paths have the same winding numbers? I think I can prove that two homotopic paths may have different winding numbers.

2. Let $C$ be a counter-clockwise loop around point $p$ in the complex plane $\Bbb{C}$. Why does the analytic image of this loop have to have a positive winding number around $f(p)$ (the image of the point under the mapping), while the continuous image may have a negative winding number?

Answer 1

To answer 2, analytic maps of $\mathbb{C}$ preserve orientation, but continuous maps such as $f(z) = \overline z$ may reverse orientation.

Answer 2

Your second question has been answered by Lee Mosher. Here is the answer to your first question:

Consider two smooth paths $$t\mapsto z_0(t), \quad t\mapsto z_1(t)\qquad(0\leq t\leq 1)$$ which both avoid the origin and are homotopic with respect to $\dot{\mathbb C}$. This means that there is a continuous map $$F: \quad [0,1]^2\to\dot{\mathbb C},\qquad (\tau,t )\mapsto z_\tau(t):=F(\tau,t)$$ with $$F(0,t)=z_0(t), \quad F(1,t)=z_1(t)\qquad(0\leq t\leq 1)\ .$$ Since the image set $S:=F\bigl([0,1]^2\bigr)\subset\dot{\mathbb C}$ is compact there exists a disk $D_\delta$ of radius $\delta>0$ such that none of the curves $t\mapsto z_\tau(t)$ meets $D_\delta$. It follows that the function $$\tau\mapsto N(\tau):={1\over 2\pi i}\int_0^1 {z_\tau'(t)\over z_\tau(t)}\>dt\ \qquad(0\leq\tau\leq1)$$ is continuous, and as it is integer-valued it has to be constant.