Proposition

Given a space $X$ and a path-connected subspace $A$ containing the basepoint $x_0$, show that the map $\pi_1(A, x_0) \rightarrow \pi_1(X, x_0)$ induced by the inclusion $A \rightarrow X$ is surjective iff every path in $X$ with endpoints in $A$ is homotopic to a path in $A$.

Solution

First, suppose that \(i_*\) is surjective. Let $p_a, p_b \in A$ and $h$ be a path in $X$ from $p_a$ to $p_b$. Since $A$ is path-connected, let $a, b$ denote paths from $x_0$ to $p_a, p_b$, respectively.

Then $a \cdot h \cdot \overline{b}$ is a loop based at $x_0$, so $[a \cdot h \cdot \overline{b}] \in \pi_1(X, x_0)$. Since \(i_*\) is surjective, there must exist a $[f] \in \pi_1(A, x_0)$ such that \(i_*([f]) = [a \cdot h \cdot \overline{b}]\). Then $[i \circ f] = [a \cdot h \cdot \overline{b}]$. This implies $[\overline{a} \cdot (i \circ f) \cdot b] = [h]$, and $\overline{a} \cdot (i \circ f) \cdot b$ is a path in $A$. Therefore, any path in $X$ with endpoints in $A$ is homotopic to a path in $A$.

Next, suppose that every path in $X$ with endpoints in $A$ is homotopic to a path in $A$. Let $[f] \in \pi_1(X, x_0)$. Then $f$ is a loop in $X$ based at $x_0$. Thus it must be homotopic to a path in $A$. Let $g$ be such a path. Then $[g] \in \pi_1(A, x_0)$, so \(i_*([g]) = [i \circ g] = [g] = [f]\), so \(i_*\) is surjective.