TITLE: Isomorphism between Homology Groups
QUESTION [5 upvotes]: Let $A$ be a retract of $X$. Show that if 
$i:A\to X$ is the inclusion operator , then $i_*:H_n(A) \to H_n(X)$ is an isomorphism
(It's easy to see that it's a monomorphism. But why is it an epimorphism?) 
Thanks!

REPLY [4 votes]: As others have mentioned, this statement is not true without a stronger condition on $r$: that it is a deformation retraction. That is, you also want $i\circ r$ to be homotopic to the identity on $X$. Then $(i\circ r)_*=(\text{id}_X)_*$ is the identity so $i_*$ is surjective. Combine this with the injectivity of $i_*$ to get that it is an isomorphism.