Working in the context of James Construction, using the following lemma:
If $X$ and $Y$ and connected CW-complexes then $$S(X \times Y) \simeq SX \vee SY \vee S(X \wedge Y)$$
and the diagram $\begin{matrix} FW_k(X)&\to &X^k &\to X^{(k)}\\\downarrow &&\downarrow & \downarrow\\ J_{K-1}(X) &\to&J_k(X) &\to X^{(K)}\end{matrix} $
where $FW_k(X)$ is the $k$th fat wedge of $X$ and $X^{(K)} = J_k(X) / J_{K-1}(X)$, I don't understand the proof of the following proposition :
Proposition: If $X$ has the homotopy type of a connected CW-complex then $$ S(J(X)) \simeq \bigvee_{k=1}^{\infty} S(X^{(k)})$$
Proof: Using the previous lemma and induction on $k$ the top right epimorphism in the preceding diagram has a homotopy splitting after suspendingm and so the bottom does as well.
I really dont understand this proof, most likely because I don't understand what the author means by "homotopy splitting".
Note : this comes from Paul Selick, Introduction to Homotopy Theory.