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.


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.


Your Answer

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

Not the answer you're looking for? Browse other questions tagged or ask your own question.