Show that if \(A\) is a retract of \(X\) then the map \(H_n(A) \to H_n(X)\) induced by the inclusion \(A \subset X\) is injective.
Solution
Let \[ i:A\hookrightarrow X \] be the inclusion and let \[ r:X\to A \] be a retraction, so \(r\circ i=\operatorname{id}_A\).
1. On homology,\[ r_*\circ i_*=\operatorname{id}_{H_n(A)}. \]
Homology is functorial, hence \[ r_*i_*=(r\circ i)_*=(\operatorname{id}_A)_*=\operatorname{id}_{H_n(A)}. \]
2. Therefore\[ \boxed{i_*:H_n(A)\to H_n(X)\text{ is injective}.} \]
If \(i_*(x)=0\), then applying \(r_*\) gives \[ x=r_*i_*(x)=0. \]