Path homotopy between loops

Question

We have two loops $f$ , $g$, one loop based at $x$ and the other at $y$ respectively. We also have a path $h$ with initial point $x$ and end point $y$. I'm trying to find a path homotopy between $f$ and $h$ * $g$ * $\overline{h}$. We also have that $f,g$ are free homotopic. Obviously $f$ and $h$ * $g$ * $\overline{h}$ are based at $x$ and I know I'm supposed to use the free homotopy condition to find the path-homotopy, but I'm totally stuck. Any hints or help would be much appreciated

Answer 1

With the clarification from the comments that $h$ is not fixed, here's a hint :

Let $H$ be a free homotopy of loops from $f$ to $g$. Consider the path $h$ from $x$ to $y$ given by $t\mapsto H(0,t)$. Do a drawing to see why that path might work, and then try to prove it.

(where $H(-,0) = f, H(-,1) = g$)

Note : if $h$ is required to be fixed, the result is wrong. For instance, take $x=y$ and $h$ to be the trivial path. Then you would be claiming that any freely homotopic loops are homotopic, which is known not to be true. That's why you have to somehow cook up an appropriate path $h$, and the free homotopy that you've assumed existed allows you to do that.

Added: here's a sketch of proof. You want to define a homotopy $H'$ based on $H$ but using the path $h$. At time $s$, $H'$ is the path that follows $h$ up until $s$, then follows $H(-,s)$, and then $h$ in the reverse direction.

You should try to make that more precise, prove that it's well-defined, continuous, and a path-homotopy between the appropriate things.