Main theorem of obstruction theory, Davis and Kirk I'm reading Davis and Kirk, in it they claim:
theorem 7.1: let $(X,A)$ be a CW pair and $Y$ is a path connected $n$-simple space with $n\geq 1$. Let $g:X_n\rightarrow Y$ be a continuous map. Then：

*

*there is a cellular cocycle $\theta\in C^{n+1}(X,A;\pi_n Y)$ which vanishes iff $g$ extends to a map $X_{n+1}\rightarrow Y$;

*the cohomology class $[g]\in H^{n+1}(X,A;\pi_nY)$ vanishes iff the restriction $g_{X_{n-1}}X_{n-1}\rightarrow Y$ extends to a map $X_{n+1}\rightarrow Y$.

My question is: what role does $A$ play here? It seems that the theorem has nothing to do with $A$ here. I suspect there is a typo in the theorem. Any help will be appreciated!
$\textbf{Oh, I see; Davis and Kirk are using the convention}$
$ \textbf{that $X_n$ means the union of $A$ and the $n$-skeleton of $A$.}$

The map is assumed to be already defined on $A$. They say a couple of lines earlier: "We wish to study the question of whether $g: A \to Y$ can be extended to map $X \to Y$". You are right that there is an inaccuracy in that the theorem itself does not mention that the function is defined on $A$.
