Hatcher 2.1.11

Hatcher, Algebraic Topology, Chapter 2, Section 1

11. 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.

Proof. Let $A\subset X$ , let $\iota:A\to X$ denote inclusion, and suppose that there exists a map $r:X\to A$ which is a retraction. In particular, then, the composition $r\iota$ is homotopic to the identity on $A$ , i.e. $r\iota\simeq 1_A$ . In terms of the induced maps, then, Theorem 2.10 ensures that $(r\iota)_*=(1_A)_*:H_n(A)\to H_n(A)$ . Utilizing the two facts on page 111 (directly beneath Proposition 2.9), it follows that $(r\iota)_*=r_*\iota_*$ and that $(1_A)_*=\mathbf{1}$ where $\mathbf{1}$ denotes the identity on the level of homology. Thus, $\mathbf{1}$ injective implies that $r_*\iota_*$ is injective, which in turn implies that $\iota_*:H_n(A)\to H_n(X)$ is injective. $\square$