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I am trying to prove a proposition relating analysis and geometry. I have a general idea on how to prove it. However, a small part of the proof needs a lemma about path homotopy and winding number. Specifically, here it is:

Let $\alpha,\beta:[0,1]\rightarrow\mathbb{C}\setminus\{p\}$ be two (continuous) paths (not necessarily closed) such that $\alpha(0)=\beta(0)$ and $\alpha(1)=\beta(1)$.

If $\alpha\simeq_\mathrm{p}\beta$, then $\mathrm{W}(\alpha,p)=\mathrm{W}(\beta,p)$.

I tried to find this theorem in books, but so far, I only found the same lemma but applied to closed curves. I found someone stating this here, but with no proof. In fact, he/she said that it can be proved easily by lifting lemma. However, I didn't take Algebraic Topology course yet for now. I have not much idea about the lifting lemma.

I really appreciate if someone could provide a proper explanation / proof / idea / reference for this lemma. Thank you in advance.

Haley13
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    What's your definition of winding number? – Dan Rust Sep 06 '14 at 23:18
  • The same as usual: $W(\alpha,p)=\frac{\theta(1)-\theta(0)}{2\pi}$, where $\theta(t)$ is angular coordinate from fixed point $p$. Usually, we take the curve to be closed, so the number is integer. But, in general, we can define it on any curve. – Haley13 Sep 07 '14 at 01:03
  • @Haley13 What definition of homotopy are you using for an arbitrary curve $[0,1] \rightarrow \Bbb C \setminus{p}$? –  Sep 07 '14 at 02:26
  • @MikeMiller A homotopy between $\alpha$ and $\beta$ is a continuous function $F:[0,1] \times [0,1] \rightarrow \mathbb{C} \setminus {p}$ such that (a) $F(0,s)=\alpha(s)$ and $F(1,s)=\beta(s)$ for all $s \in [0,1]$ (b) $F(t,0)=\alpha(0)=\beta(0)$ and $F(t,1)=\alpha(1)=\beta(1)$. – Haley13 Sep 07 '14 at 20:52
  • I forgot to add extra assumption that $\alpha(0)=\beta(0)$ and $\alpha(1)=\beta(1)$. – Haley13 Sep 07 '14 at 20:53
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    Mm - what I said wasn't quite true. (I misunderstood what you meant by angle function.) I've deleted my comment. I don't have any good suggestions for a textbook, but if you can find the version for closed curves, it might be a good exercise to extend it yourself - the argument should be similar. –  Sep 07 '14 at 21:02
  • Relevant: http://math.stackexchange.com/questions/112679/why-is-the-winding-number-homotopy-invariant. Also see http://www.math.uiuc.edu/~mando/lectures/430/0901.pdf –  Sep 07 '14 at 21:05
  • Massey is the best book I know for this thing. – Moishe Kohan Sep 07 '14 at 21:16

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