Formula for relative homotopy groups of products

Question

In Hatcher prop 4.2 he proves that the n-th homotopy group of a product $X\times Y$ (for $X$ and $Y$ path-connected) is isomorphic to the product of the n-th homotopy groups of $X$ and $Y$. I wonder if a similar statement is true for relative homotopy groups? I.e. do we have a statement like $\pi_n(X\times Y, A\times B, (x_0,y_0))=\pi_n(X,A,x_0)\times \pi_n(Y,B,y_0)$?

Answer 1

$\pi_n(Y,B,b_0)$ is the set of homotopy classes of maps $(D^n,S^{n-1},*) \to (Y,B,b_0)$. Hatcher's proof of 4.2 applies verbatim to relative homotopy groups: Maps $(D^n,S^{n-1},*) \to (X_ 1\times X_2, A_1 \times A_2, (a_1,a_2))$ can be identified with pairs of maps $(D^n,S^{n-1},*) \to (X_ i, A_i,a_i)$, and the same is true for homotopies.