In Hatcher, Corollary 3A.4 stated a universal coefficient theorem for relative homology, i.e. the following short exact sequence splits:
$0 \rightarrow H_n(X,A) \otimes_\mathbb{Z} G \rightarrow H_n(X,A;G) \rightarrow Tor_\mathbb{Z}^1(H_{n-1}(X,A),G)\rightarrow 0$
, where $G$ is an abelian group.
Since $S_n(X,A)$ is a subquotient of $S_n(X)$, the chain complex $S_\cdot (X,A)$ does not consist of free $\mathbb{Z}$-modules and the proof of universal coefficient theorem for homology group does not work here. How can I solve this problem?
Best Answer
$S_n(X, A)$ is a free $\mathbb Z$ module on the simplices $\Delta^n \to X$ whose image is not contained in $A$.