[Math] Equal winding number implies two paths are path homotopic

Question

Let $\alpha,\beta:[0,1]\rightarrow\mathbb{C}\setminus\{p\}$ be two (continuous) paths (not necessarily closed) with same endpoints ($\alpha(0)=\beta(0)$, $\alpha(1)=\beta(1)$), we know that if $\alpha\simeq_\mathrm{p}\beta$, then $\mathrm{W}(\alpha,p)=\mathrm{W}(\beta,p)$ by using lifting lemma. However, is the converse also true? If so, how to construct a homotopy?

Answer 1

OK, I found a possible proof:

Let $\overline{\alpha}$, $\overline{\beta}$ be the lifting of $\alpha$, $\beta$ such that $\overline{\alpha}(0)=\overline{\beta}(0)$, then $\mathrm{W}(\alpha,p)=\mathrm{W}(\beta,p)$ implies there is a path homotopy between $\overline{\alpha}$, $\overline{\beta}$, (both coordinate use path homotopy since they are paths in $\mathbb{R}_{>0}\times\mathbb{R}$). Then composite with function $(r,\theta)\mapsto r\exp(i\theta)$ would be a path homotopy between $\alpha$ and $\beta$.