# $\pi_1$ action on relative homotopy groups $\pi_n(X,A)$

It is known that there is a natural $$\pi_1(X,x)$$ action on $$\pi_n(X,x)$$ which also induces a bijection $$\pi_n(X,x)/\pi_1(X,x) \cong [S^n, X]$$.

Now, let $$(X,A)$$ be a pair of path-connected spaces and $$x\in A$$. We also have a $$\pi_1(A,x)$$ action on the relative homotopy group $$\pi_n(X,A,x)$$.

Is there any similar description for the quotient $$\pi_n(X,A,x)/ \pi_1(A,x)$$?

I guess it might be $$[(D^n,S^{n-1}), (X,A)]$$, the homotopy class of continuous maps of space pairs, since I notice that $$\pi_n(X,A, x)$$ can be defined as $$[(D^n, S^{n-1},pt), (X,A,x)]$$.

But maybe I was wrong, as I cannot find it in any reference.

Let's consider the general case of maps $$(X,A,x_0) \rightarrow (Y,B,y_0)$$ with $$Y$$ and $$B$$ path-connected. To be explicit the action is the following: given a loop $$\gamma$$ in $$B$$, it acts on $$[f] \in \langle (X,A,x_0),(Y,B,y_0)\rangle$$ by sending it to a class of a relative map $$g$$ that is homotopic to $$f$$ through a relative homotopy that moves the basepoint along $$\gamma$$.
We show that $$\langle (X,A,x_0), (Y,B,y_0) \rangle / \pi_1(Y,y_0) \rightarrow [(X,A),(Y,B)]$$ is a bijection:
It is surjective because given a relative map $$f$$, we can find a path from $$f(x_0)$$ to $$y_0$$ in $$B$$ which we can extend to a homotopy $$A \times I \rightarrow B$$ which we can extend to a homotopy $$X \times I\rightarrow Y$$.
It is injective by definition of the action since two classes that map to the same element are represented by maps that are homotopic through a relative homotopy taking a basepoint along a loop in $$B$$.
So taking $$X=(D^n,S^{n-1})$$ you get the result.