long exact sequence in $K$-theory

Question

I am studying the basics of $K$-theory and given a CW pair $(X,A)$ I understand how to construct an long exact exact sequence

$\cdots \rightarrow K(SX) \rightarrow K(SA) \rightarrow K(X/A) \rightarrow K(X) \rightarrow K(A)$.

I many of the examples I encounter, the inclusion map $A \rightarrow X$ has a splitting $X \rightarrow A$ making $K(X) \rightarrow K(A)$ surjective. Does this imply automatically that $K(X/A) \rightarrow K(X)$ is injective? Yes, no, why? Thanks.

Answer 1

If $A \to X$ is a retract, then also $SA \to SX$ is a retract. Then $K(SX) \to K(SA)$ is surjective, which means that $K(X/A) \to K(X)$ is injective.