Splitting of a spectrum as a wedge Suppose that $i:E\to F$ and $r:F\to E$ are maps of spectra ($S^1$-spectra of topological spaces) such that $r\circ i$ is a homotopy equivalence. Can we always show that the spectrum $F$ splits as a widge of $E$ and the cofibre of $i$? Several authors seem to be using such splitting without further explanation, so apologies if the answer depends on some standard results or constructions, but I am not very familiar with the stable homotopy theory.

Yes, this is true. Let $C$ denote the cofiber of $i$, and let $T$ be any test spectrum. Then using the retract $r$, we obtain a split short exact sequence $$0 \to [T,E] \to [T,F] \to [T,C] \to 0.$$ Thus we have natural isomorphisms $$[T,F] \cong [T,E] \oplus [T,C] \cong [T, E \vee C].$$ So we have $F \simeq E \vee C$ by the Yoneda lemma.