$K$-theory exact sequence. Let $Y$ be a closed subspace of a compact space $X$. Let $i:Y \to X$ the inclusion and $r:X \to Y$ a retraction ($r \circ i = Id_Y$). I have to prove that exists this short exact sequence
$$ 0 \to  K(X,Y) \to K(X) \to K(Y)  \to 0.$$
Then I have to verify that $K(X) \simeq K(X,Y) \oplus K(Y)$. How can I do it? I think that $K(X,Y) = \tilde{K}(X/Y).$ Thank you very much.

For the exactness at the left, you can use the long exact sequence in $K$ theory. Then every map involving (suspensions) of $i$ will split (why). Then the long exact sequence breaks down in short exact sequences.
