Proving that $H_{n}(X,A)$ is isomorphic to $\widetilde{H}_{n}(X)$ if A is acyclic.

Question

I encountered this exercise in Bredon's book "Topology and Geometry". I managed to prove the case where $n>1$ using the exact sequence of the pair $(X,A)$ and the property that relative homology is isomorphic to homology of the same space for dimension higher than 1 but I am not sure how to continue. I would prefer hints over the full answer please.

Thank you!

Answer 1

Just consider the fragments $$\cdots \to\widetilde{H}_n(A) \to \widetilde{H}_n(X) \to H_n(X,A) \to \widetilde{H}_{n-1}(A) \to \cdots $$ of the (reduced) long exact sequence of the pair $(X,A)$. Bredon mentions the reduced version of the long exact sequence in page 185.

Answer 2

There is a map of pairs $(X,\ast)\to (X,A)$ which induces a map between long exact sequences of pairs. The maps associated to $1:X\to X$ and to $\ast\to A$ are isomorphisms. Now use the 5 lemma to conclude that $H_*(X,A)$ is isomorphic to $H_*(X,\ast)=\widetilde H_*(X).$