Hatcher-Algebraic-Topology · Problem 4.1.7

The key insight is that basepoint changes in relative homotopy groups $\pi_n(X, A, x_0)$ follow the same pattern as absolute ones, but now you're tracking homotopy classes of pairs $(f, F)$ where $f$ maps the $n$-sphere relative to its basepoint, and $F$ is a homotopy that respects the subspace $A$. The induced map $\beta_\gamma$ acts by conjugation with a path $\gamma$ in $X$, and the proof mirrors the absolute case—you just need to verify that the homotopy $F$ stays within $A$ throughout, making the relative structure transparent to the basepoint-change argument.

The key insight is that basepoint changes in relative homotopy groups $\pi_n(X, A, x_0)$ follow the same pattern as absolute ones, but now you're tracking homotopy classes of pairs $(f, F)$ where $f$ maps the $n$-sphere relative to its basepoint, and $F$ is a homotopy that respects the subspace $A$. The induced map $\beta_\gamma$ acts by conjugation with a path $\gamma$ in $X$, and the proof mirrors the absolute case—you just need to verify that the homotopy $F$ stays within $A$ throughout, making the relative structure transparent to the basepoint-change argument.