Determining images of points in a path homotopy.

Question

Say the two paths $f_0$ and $f_1$ ae homotopic. Then $(1-t)f_0+tf_1$ is the homotopy between the two paths.

Say $f_0,f_1\in\Bbb{R^2}$, and there is a point $(a,b)$ in $f_0$. How can we find which point in $f_1$ is $(a,b)$ mapped to? Is there a unique mapping for each point in $f_0$?

Thanks.

Answer 1

I suppose you want to know where $(a,b)$ travels during the homotopy from $f_0$ to $f_1$? If $f_0$ is an injective path, so it has no self-intersections, the answer is $f_1(f_0^{-1}(a,b))$.

If $f_0$ is not injective, $(a,b)$ might have more than one preimage and theres no unique answer. Consider the following homotopic curves: