Show that the image of the map \(\pi_2(X, x_0) \to \pi_2(X, A, x_0)\) lies in the center of \(\pi_2(X, A, x_0)\).
Solution
Let \[ j_*:\pi_2(X,x_0)\to\pi_2(X,A,x_0) \] be the natural map, and let \[ a\in\operatorname{im}j_*. \] Exactness of the long homotopy sequence of the pair gives \[ \partial a=0\in\pi_1(A,x_0). \] For arbitrary \(b\in\pi_2(X,A,x_0)\), Lemma 4.39 says \[ a+b-a=(\partial a)b. \] Since \(\partial a\) is the identity element of \(\pi_1(A,x_0)\), its action on \(b\) is trivial, so \[ a+b-a=b. \] Hence \[ a+b=b+a. \] Thus every element in the image of \(j_*\) commutes with every element of the relative group: \[ \boxed{\operatorname{im}\bigl(\pi_2(X)\to\pi_2(X,A)\bigr) \subseteq Z\bigl(\pi_2(X,A)\bigr).} \]