Degree, Angle, Period and maps onto the unit circle This question is a follow-up from this question, whose accepted answer was to go and read up Allen Hatcher's book on Algebraic Topology, Chapter 1. I read it, but it only helps up to a point.
The question is: suppose that $\Omega=D_2\setminus \overline{D_1}=\{(x_1,x_2) : 1< x_1^2+x_2^2 <4\}$, and let $\phi$ be a smooth map from $\Omega$ to $S^1=\{(x_1,x_2) : 1= x_1^2+x_2^2\}$. Can I write
$$
\phi = \left(\cos(\theta),\sin(\theta)\right),
$$
for some function $\theta\in C^\infty(\Omega;\mathbb{R}/(2\pi k\mathbb{Z}))$, for some $k$ possibly depending on $\phi$? And if not, what can I write?
I am doing a bit of cargo cult mathematics here, because I think this is what I read means, but am certainly not confident and I don't master the machinery. If $\phi$ was a map from $S^1$ to $S^1$, then I think this is what it would be, and $k$ would be the Brouwer degree of $\phi$. And since $\Omega$ is $2$ dimensional but homotopic to the circle, it should be the same...but is that true?
Going further, I can understand functions $ C^\infty(\Omega;\mathbb{R}/(2\pi\mathbb{Z}))$. So is it the case that, in fact, degree has nothing to do with it and the answer is simply
$$
\theta\in C^\infty(\Omega;\mathbb{R}/(2\pi \mathbb{Z}))
$$
(and Algebraic Topology says something simple, with no $k$ index subgroup involved..)?
 A: Not quite sure where your problem lies, but here is a fairly constructive way to proceed:
Suppose you remove, say $C=\{(x_1,0): 1<x_1<2\}$ from $\Omega$. Then $\Omega^*=\Omega\setminus C$ is simply connected and $\phi:\Omega^*\to S¹$ lifts to a well-defined map $\theta: \Omega^* \to {\Bbb R}$ . This $\theta$ admits limits $\theta_+: C \to {\Bbb R}$ and $\theta_-: C \to {\Bbb R}$ (from above and from below the cut) and by continuity we must have $\theta_+-\theta_-= 2\pi k$ for some $k\in {\Bbb Z}$, called the degree of the map $\phi$.
It corresponds to what happens to the lift of a generating loop $\gamma$ in $\Omega$. $k$ is called the degree of the map $\phi$.
Any loop $\alpha$ in $\Omega$ is homotopic to a multiple, say $m\in {\Bbb Z}$, of $\gamma$ so going around $\alpha$ will lead to a difference $2\pi k m$ when lifting. Thus, we obtain a well-defined map $\widehat{\theta} : \Omega \to {\Bbb R}/(2\pi k {\Bbb Z})$. When $k=0$ then you simply get a well defined function with values in ${\Bbb R}$. You may always compose with a projection to obtain a map with values in ${\Bbb R}/(2\pi {\Bbb Z})$ but you loose information this way.
