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 ?

  • $\begingroup$ 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
  • 3
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.