# Checking homotopy of curves

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. 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.

## 1 Answer

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.