In the last paragraph in page 345 of Hatcher's Algebraic Topology(link:http://pi.math.cornell.edu/~hatcher/AT/ATch4.pdf), Hatcher says that $\pi_1(A,x_0)$ acts on the long exact sequence of homotopy groups for $(X,A,x_0)$, the action commuting with the various maps in the sequence.
I can't see the commutativity.
For $[f] \in \pi_n(X,x_0)$, the action is defined by $[\gamma][f]=[\gamma f]$ where $\gamma f$ is the map as in the following figure (on the left), while for $[f] \in \pi_n(X,A,x_0)$, $\gamma f$ is defined as in the right figure. Also, these two are not homotopic as maps $(I^n,\partial I^n,J^{n-1})\to (X,A,x_0)$ in gerenal. Then how can the action commute with the map $\pi_n(X,x_0) \to \pi_n(X,A,x_0)$?
Best Answer
I think all actions of $\pi_1(A,x_0)$ are the relative ones.
In particular, I think Hatcher sees the long exact sequence of $(X,A,x_0)$ as follows :
$\cdots\to\pi_n(A,x_0,x_0)\to\pi_n(X,x_0,x_0)\to\pi_n(X,A,x_0)\to\pi_{n-1}(A,x_0,x_0)\to\cdots$
This seems to be consistent with the proof of Theorem 4.3. Thus it makes sense to interpret all actions of $\pi_1(A,x_0)$ to any of the groups above by the right figure in your post.
The commutativity of the action of $\pi_1(A,x_0)$ should be clear if we stick with the relative version.