Given two maps $f,g : S^1 \rightarrow S^1$, I want to show that if they have the same winding number, then there is a homotopy between them.
Well, we know that we can write them as $f(\exp(2 \pi i t))= f(1) \exp(2 \pi i \phi_f(t))$ and $g(\exp(2 \pi i t))= g(1) \exp(2 \pi i \phi_g(t))$. Where $\phi_f(0) = \phi_g(0) = 0$ and $\phi_f(1) = \phi_g(1) = n$, where $n$ is the winding number.
Now my first idea was that with $z(t):=\phi_f(t)-\phi_g(t)$ and $g(1)/f(1) = \exp(2 \pi i k)$ for some constant $k$, that $H(t,s) = f(\exp(2 \pi i t)) \cdot \exp(2 \pi i k s) \cdot \exp(2 \pi i s z(t))$ should do it, but then I was wondering: Where exactly did I use that they have the same winding number? So my proof must be wrong. Can andybody here correct it?