$\pi_1(A,x_0)$ acts on the long exact sequence of homotopy groups for $(X,A,x_0)$

Question

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 $\pi_n(X,x_0) \to \pi_n(X,A,x_0)$?

Answer 1

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.

Answer 2

This question has another another approach to an answer which is taken in Section 6.3 of the book Nonabeian Algebraic Topology {NAT}. The basic idea is to form for a triple $(X,A,C)$ of spaces, i.e. $C \subseteq A \subseteq X$, and $C$ is thought of as a set of base points, a functor $\rho(X,A,C)$ which which in dimension $0$ is $C$; in dimension $1$ is $\pi_1(A,C)$, the fundamental groupoid of $A$ on a set of base points; and in dimension $2$ is the set of homotopy classes rel vertices of maps $$(I^2, \partial I^2, \partial \partial I^2) \to (X,A.C), $$where $\partial I^2$ is the boundary of the square $I^2$ and $\partial \partial I^2$ is the set of vertices of the square. You will notice that this structure does not form a group!

However it turns out that for each $c \in C$ the obvious map obvious map $\pi_2(X,A,c) \to \rho_2(X,A,C)$ is injective (Proposition 6.3.8). Further $\rho_2(X,A,C)$ has two compositions $+_1, +_2$ which make the whole structure into a double groupoid . There is also an extra structure of so-called connections which make it equivalent to the crossed module $$\partial: \pi_2(X,A,C)\to \pi_1(A,C).$$

Having got all this done Chapter 6 of NAT goes on to prove a 2-d Seifert Van Kampen Theorem for $\rho$ which by translation to the equivalent crossed module allows new calculations of 2nd relative homotopy groups, as pushouts of crossed modules.

The point is that the additional structures allow, once you have mastered them, clearer and more powerful proofs. For more discussion of the history and methodology, see this paper.