Show that if \(A\) is a path-connected subspace of \(X\), \(x_0\in A\), and the homomorphism \(i_*:\pi_1(A,x_0)\to\pi_1(X,x_0)\) induced by the inclusion map \(i:A\to X\) is surjective, then every path in \(X\) with endpoints in \(A\) is homotopic relative to the endpoints (i.e., via a homotopy that fixes each endpoint) to a path in \(A\).
Solution
Let \[ \alpha:[0,1]\longrightarrow X \] be a path with \[ \alpha(0)=a\in A, \qquad \alpha(1)=b\in A. \]
1. There are paths\[ \lambda:[0,1]\to A, \qquad \mu:[0,1]\to A \]with\[ \lambda(0)=x_0, \quad \lambda(1)=a, \qquad \mu(0)=b, \quad \mu(1)=x_0. \]
The points \(x_0,a,b\) all lie in the path-connected space \(A\). Hence there is a path in \(A\) from \(x_0\) to \(a\) and a path in \(A\) from \(b\) to \(x_0\). Choose these as \(\lambda\) and \(\mu\).
2. The concatenation\[ \ell=\lambda*\alpha*\mu \]is a loop in \(X\) based at \(x_0\), and there is a loop\[ \gamma:[0,1]\to A \]based at \(x_0\) such that, as loops in \(X\),\[ \gamma\simeq\ell \quad\text{relative to the basepoint}. \]
The endpoint equalities in 1 and those for \(\alpha\) make the concatenation well-defined and give \[ \ell(0)=\ell(1)=x_0. \] Thus \([\ell]\in\pi_1(X,x_0)\).
Surjectivity of \[ i_*:\pi_1(A,x_0)\to\pi_1(X,x_0) \] gives a class \([\gamma]\in\pi_1(A,x_0)\) satisfying \[ i_*[\gamma]=[\ell]. \] Choose a loop \(\gamma\) in \(A\) representing this class. Equality of the two classes in \(\pi_1(X,x_0)\) means precisely that, after viewing \(\gamma\) as a loop in \(X\), it is homotopic to \(\ell\) through loops based at \(x_0\).
3. The path\[ \beta=\lambda^{-1}*\gamma*\mu^{-1} \]lies entirely in \(A\) and runs from \(a\) to \(b\).
The reversed path \(\lambda^{-1}\) runs in \(A\) from \(a\) to \(x_0\), the loop \(\gamma\) lies in \(A\) and begins and ends at \(x_0\), and \(\mu^{-1}\) runs in \(A\) from \(x_0\) to \(b\). Hence their concatenation is an \(A\)-valued path from \(a\) to \(b\).
4. For any path \(\eta\) from \(u\) to \(v\), the loops\[ \eta^{-1}*\eta \quad\text{at }v \qquad\text{and}\qquad \eta*\eta^{-1} \quad\text{at }u \]are homotopic relative to their endpoints to the corresponding constant loops.
With the usual equal-speed parametrization, \[ (\eta^{-1}*\eta)(t)=\eta(|2t-1|). \] The formula \[ H(t,s)=\eta\bigl(s+(1-s)|2t-1|\bigr) \] is a continuous homotopy from \(\eta^{-1}*\eta\) at \(s=0\) to the constant loop at \(v=\eta(1)\) at \(s=1\), and \[ H(0,s)=H(1,s)=v \] for all \(s\).
Similarly, \[ (\eta*\eta^{-1})(t)=\eta(1-|2t-1|), \] and \[ K(t,s)=\eta\bigl((1-s)(1-|2t-1|)\bigr) \] contracts it relative to its endpoints to the constant loop at \(u=\eta(0)\).
5. The paths \(\alpha\) and \(\beta\) are homotopic relative to the endpoints \(a,b\).
By 2, concatenate the based homotopy \[ \gamma\simeq\lambda*\alpha*\mu \] on the left with the fixed path \(\lambda^{-1}\) and on the right with the fixed path \(\mu^{-1}\). This gives a homotopy relative to \(a,b\) from \[ \beta = \lambda^{-1}*\gamma*\mu^{-1} \] to \[ \lambda^{-1}*(\lambda*\alpha*\mu)*\mu^{-1}. \] The two parenthesizations of a fixed finite concatenation traverse the same component paths in the same order and differ only by an increasing piecewise-linear reparametrization \(\varphi:[0,1]\to[0,1]\) fixing \(0\) and \(1\). The straight-line interpolation \[ \varphi_s(t)=(1-s)\varphi(t)+st \] also fixes \(0,1\), so composing the path with \(\varphi_s\) gives a homotopy relative to endpoints between the two parametrizations. Consequently the latter path is homotopic relative to \(a,b\) to \[ (\lambda^{-1}*\lambda)*\alpha*(\mu*\mu^{-1}). \] By 4, the first parenthesized loop contracts at \(a\) and the second contracts at \(b\), both with their endpoints fixed. Concatenating those contractions with the fixed path \(\alpha\) yields a homotopy relative to \(a,b\) from the displayed path to \(\alpha\). Therefore \[ \beta\simeq\alpha\pmod{\text{endpoints}}. \] By 3, \(\beta\) lies in \(A\), which proves the claim.