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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?

RobPratt
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Kaa1el
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    First, the winding number is not defined for non-closed paths, you need a loop for this. Second, you have to learn about the fundamental group of the circle and how to prove that it is isomorphic to ${\mathbb Z}$. – Moishe Kohan Jan 30 '14 at 23:21
  • No, the winding number can be defined for any path by lifting with respect to polar coordinate function; and for closed loop, it is indeed an integer. – Kaa1el Jan 30 '14 at 23:27
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    Then you would surely need the added condition that $\alpha(0)=\beta(0)$ and $\alpha(1)=\beta(1)$ and then talk about homotopies relative to end points. – Dan Rust Jan 30 '14 at 23:29
  • yes, you're correct, my mistake:) – Kaa1el Jan 30 '14 at 23:31

1 Answers1

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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 $\beta$.

Kaa1el
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