let S be the nsphere. how can we see that
(SxS) smash S
has the homotopy type of a wedge of spheres?
let S be the nsphere. how can we see that (SxS) smash S has the homotopy type of a wedge of spheres? 


More generally the suspension $S\land(X\times Y)$ of $X\times Y$ is the wedge of the suspensions of $X$, $Y$, and the smash product $X\land Y$; see for example proposition 4.74 of Hatcher's book. 

