Why is $S_{\ast}\left(X,A\right)$ free? it is the quotient of two free groups $S_{\ast}\left(X\right)$ & $S_{\ast}\left(A\right)$
2026-04-02 16:28:48.1775147328
Why is $S_{\ast}\left(X,A\right)$ free?
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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.