# when are the winding number of homotopic curves not equal?

For what cases do we have it that even though two closed rectifiable curves are homotopic, that they do not share the same winding number ?

• In the plane, any two closed curves are homotopic, regardless of winding number. Is that what you're looking for? – Arthur Oct 25 '18 at 22:09
• @Arthur no not quite , there is a theorem which states that for $w \in \Bbb C-G$ and $\gamma_0$ homotopic to $\gamma_1$ then $n(\gamma_1,w)=n(\gamma_0,w)$. But consider $\gamma_0=e^{2\pi it},\gamma_1=e^{4pi i t}$. these two curves are homotopic in $\Bbb C$ but their winding numbers around zero are different. why is this ? – Voltron Oct 25 '18 at 22:14
• If you want homotopy classes of curves to be the same thing as winding numbers about a point $p \in \mathbb{C}$, you must consider homotopies in $\mathbb{C} \setminus \{p\}$, the punctured plane. – Joppy Oct 25 '18 at 23:31