For loops in C times, the winding number about 0 equals the circle degree

Because $\gamma (t)\mathrm{\ne }0$ for every $t$, the normalized map $\alpha (t)=\gamma (t)\mathrm{/}\mathrm{\mid }\gamma (t)\mathrm{\mid }$ is a continuous loop in the unit circle based at $1$. Using the homeomorphism of [L2], regard the same loop as a based loop $\beta :[0,1]\to \mathbb{R}\mathrm{/}\mathbb{Z}$ at $[0]$. Let $\tilde{\beta }:[0,1]\to \mathbb{R}$ be its lift with $\tilde{\beta }(0)=0$, and define $\theta (t)=2\pi \tilde{\beta }(t)$.

By the definition of $h$ in [L2], the lift from step 1.1 supplies a continuous argument. [step 1.1, L1, L2, algebra] $\alpha (t)=\cos \theta (t)+i\sin \theta (t)=e^{i\theta (t)}\mathrm{.}$ Hence $\gamma (t)=\mathrm{\mid }\gamma (t)\mathrm{\mid }{e}^{i\theta (t)},$ so $\theta$ is a continuous argument of $\gamma$ about $0$. Therefore [L1] gives $n(\gamma ,0)=\frac{\theta (1)-\theta (0)}{2\pi }=\tilde{\beta }(1)\mathrm{.}$

Since $\tilde{\beta }(1)$ is exactly the degree of $\beta$ by [L3], step 2.1 shows $n(\gamma ,0)=\mathrm{deg}(\beta )$, which is the asserted degree of the normalized circle loop of $\gamma$. Fact [L4] records that this is the same integer that classifies the loop class in the usual ${\pi }_1(S^1)\cong \mathbb{Z}$ convention.