This question is similar but different from this Let the spaces be locally compact, Hausdorff and based.
Problem: Construct a map $\Sigma (X \times Y) \to \Sigma X \vee \Sigma Y \vee \Sigma (X \wedge Y)$ which induces isomorphism on homology groups.
I think once again I must use the fact that $[\Sigma Z, W]_*$ has a group structure.
We do get the natural map,
$\Sigma (X \times Y) = \Sigma X \vee \Sigma Y \vee \Sigma (X \wedge Y)$ (where the first two are induced by projections and the last one by quotient)
and assuming $(X \times Y, X \vee Y)$ is a good pair and using $H_{k+1}\Sigma = H_{k} $ we get the following long exact sequence,
$\cdots \to H_{k+1}(X \times Y, X \vee Y ) \to H_{k}(X \vee Y) \to H_{k}(X \times Y) \to H_{k}(X \times Y, X \vee Y) \to H_{k-1}(X \vee Y)\to \cdots$
Now I want to have that the middle portion is split sicne that will give me $H_{k}(X \vee Y) \oplus H_{k}(X \times Y, X \wedge Y) \cong H_{k}(X \times Y)$ but sice no other conditions are given I cannot conclude that.
Can anyone give me some hints?