I understand what is meant by the degree of a continuous map $f$ from $S^1$ to $S^1$. If we let $[S^1, S^1]$ denote the set of homotopy classes of continuous maps from $S^1$ to $S^1$, it turns out that the degree map gives a bijection from $[S^1, S^1]$ to the integers. I am also cool with this.
My problem is I heard that this bijection fact is equivalent to the following:
The degree map from $C(S^1,S^1)$ to the integers is a continuous map, and whenever $deg(f_0) = deg(f_1)$, there exists a path in $C(S^1,S^1)$ from $f_0$ to $f_1$.
How are the two notions equivalent? I don't have a very good grasp (or intuition) for continuous maps from $C(S^1,S^1)$ to the integers, and how that is related to modding functions out by homotopy.
(You may assume that I have background knowledge equivalent to Munkres chapter 9)
Best Answer
If you lift the map $f\colon S^1\to S^1$ to a map $\tilde f\colon \Bbb R\to\Bbb R$, then the degree is given by $\dfrac{\tilde f(2\pi)-\tilde f(0)}{2\pi}$, and you can easily construct a path joining two maps of the same degree by taking the straight-line homotopy between their respective lifts.