Why is $S_{\ast}\left(X,A\right)$ free? Why is $S_{\ast}\left(X,A\right)$ free? it is the quotient of two free groups $S_{\ast}\left(X\right)$ & $S_{\ast}\left(A\right)$

The quotient  is free because the smaller group is generated  by a subset of a basis of the larger one.
Indeed, SX is freely generated by all singular simplices in X, and SA is generated by the set of simplices in X whose image contained in A.
