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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 ?

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  • $\begingroup$ In the plane, any two closed curves are homotopic, regardless of winding number. Is that what you're looking for? $\endgroup$ – Arthur Oct 25 '18 at 22:09
  • $\begingroup$ @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 ? $\endgroup$ – Voltron Oct 25 '18 at 22:14
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    $\begingroup$ 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. $\endgroup$ – Joppy Oct 25 '18 at 23:31

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