Given a space $X$ and a path-connected subspace $A$ containing the basepoint $x_0$, show that the map $\pi_1(A,x_0) \to \pi_1(X,x_0)$ induced by the inclusion $A \to X$ is surjective iff every path in $X$ with endpoints in $A$ is homotopic to a path in $A$.
I am reading a solution in which for the "$\Longleftarrow$" direction they state that if $[f] \in \pi_1(X,x_0)$ is a loop based at $x_0$, then by assumption $f$ is homotopic to a path say $g$ in $A$. Now $g \in \pi_1(A,x_0)$ and so if $\iota_*: \pi_1(A,x_0) \to \pi_1(X,x_0)$ is the map induced by the inclusion we have that $\iota_*([g])=[\iota \circ g]= [g]=[f]$.
There are few questions which I arose here.
- I think they assume without stating that $g$ is actually a loop based at $x_0$ and not just arbitary path?
- If $[f] \in \pi_1(X,x_0)$, then if we are talking about the loop $f$ being homotopic to some other loop say $g$ we must have that $g$ is also a loop based at $x_0$ as otherwise they wouldn't be homotopic?
Best Answer
"Homotopy of paths" means that the endpoints are kept fixed. Allowing free homotopies (which can move the endpoints) does not make sense. In fact, any two paths in the same path component of a space are freely homotopic.
Therefore yes, $g$ is a loop based at $x_0$.
Yes. We have to take homotopy of paths.