I'm sure this isn't too difficult but i can't seem to do it
if you have two loops $p_0 = e*g $ and $p_1 = g*e$ where $e$ is the trivial loop
How would i construct an explicit homotopy between the two, obviously they're homotopic and are end point preserving but what would the actual $\text{homotopy} = F(x,t)$ look like
thank you !
We show that $g$ is homotopic to $g*e$, by defining $H:[0,1]\times[0,1]\to X$ by$$H(x,t)=\left\{\begin{array}{rl}g((t+1)x)&0\leq x\leq\frac{1}{t+1}\\g(1)&\frac{1}{t+1}\leq x\leq1\end{array}\right..$$It is easy to define a similar homotopy between $g$ and $e*g$. Then, the two homotopies can be joined to one homotopy.