Let $X $ be a topological space and $\mathbb {S}^1$ be the set of complex numbers with magnitude 1 equipped with the inherited topology from $\mathbb {C} $. If we have 2 loops $f,g:\mathbb {S}^1\rightarrow X $ such that $f (1)=g (1) $ and $f$ is homotopic to $g $ relative to $\emptyset $.
Must $f,g $ be homotopic relative to $\{1\} $?
Thank you.
no, not necessary.
"$f,g $ are homotopic relative to $\{1\} $" is equivalent to "$[f]=[g]$ in $\pi_1(X)$".
"$f$ is homotopic to $g $ relative to $\emptyset$" is equivalent to "$[f]$ and $[g]$ lay in the same conjugacy class".
it is easy to construct an example for arbitrary $X$ with nonabelian $\pi_1$.