Given a space $X$ and a path-connected subspace $A$ containing the basepoint $x_0$, show that the map $π_1(A, x_0) \to π_1(X, x_0)$ induced by the inclusion $A \hookrightarrow X$ is surjective iff every path in $X$ with endpoints in $A$ is homotopic to a path in $A$.
I'm wondering the proof of reverse direction. Most of the proof says that as every path in $X$ with endpoints in $A$ is homotopic to a path in $A$, if we choose a loop in $X$ based at $x_0$, then is homotopic to $\textbf{a loop in $A$ based at $x_0$}$. I don't understand this. Hypothesis only assume it's homotopic to some path in $A$ so I think the homotopic path need not be a loop. Am I right or I'm misunderstanding something here? In this case how can I prove?
Best Answer
In the context of fundamental groups "homotopic" always means "path homotopic", i.e. homotopies are required to keep the endpoints of the paths fixed.
Understanding it in the sense of arbitrary homotopies does not make because any two paths in the same path component of $X$ are homotopic.