This question concerns the well-known homotopy equivalence $$ \Sigma (X\times Y) \simeq \Sigma (X \vee \ Y) \vee \Sigma (X\wedge Y) $$ (I'm happy to use only CW complexes). I can see that there is a partition $$ \Sigma (X\times Y) = \Sigma (X\vee Y) \cup C^\circ ( X\wedge Y) $$ on the point-set level (here $C^\circ A$ denotes the open cone on the space $A$). This is consistent with (but does not imply that) $$ \Sigma (X\times Y)\quad \mbox{is homeomorphic to}\quad \Sigma (X\vee Y)\cup_\alpha C( X\wedge Y), $$ the mapping cone of some map $X\wedge Y \to \Sigma (X\vee Y)$ for some map $\alpha: X\wedge Y$ (which is necessarily nullhomotopic).
My question is: Is there such a map $\alpha$? Is it natural?
