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Consider two curves defined as $f:S^1 \rightarrow \mathbb{R}^2-\left\lbrace x_0\right\rbrace$ and $g:S^1 \rightarrow \mathbb{R}^2-\left\lbrace x_0\right\rbrace$. You can see them in the picture below.

f and g

I need to answer the following questions:

  1. Are they homotopic?
  2. Keeping $x_0$ fixed for f, determine all possible regions L of $\mathbb{R}^2 - g(S^1)$ such that, taking $x_0\in L$, f and g are homotopic.

For the first one, I noticed their winding numbers, with respect to $x_0$, are different, so, $f$ and $g$ are not homotopic.

I'm not sure how to proceed with the second question. I have the feeling I may have not been given enough information in class to answer that. Homotopy was defined very intuitevely: two curves are homotopic if they can be deformed into each other continuously.

I searched for a better definition to see if it helped (it didn't): $f_1:X\rightarrow Y$ and $f_2: X \rightarrow Y$ are homotopic if there exists a continous map $F:X\times [0,1] \rightarrow Y$ such that $F(x,0)=f_0(x)$ and $F(x,1)=f_1(x)$.

Just a nudge in the right direction would be greatly appreciated.

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The wording of question 2 is very poor. My best guess is that they are asking you consider each of the six different regions $L$ of $\mathbb R^2 - g(S^1)$, and to imagine that $x_0$ is placed, in turn, in each of those regions $L$, and then to reconsider question 1 for each of those six placements.

Assuming this interpretation is correct, you can certainly make progress on this question. You've already computed the winding number of $f$ around $x_0$. Next you can compute the six winding numbers of $g$, around points in the six different regions $L$. Once you've done those computations, perhaps you can make further progress.

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