# the homotopy type of a product of some spaces

let S be the n-sphere. how can we see that

(SxS) smash S

has the homotopy type of a wedge of spheres?

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Is this homework? – Fernando Muro Mar 29 '11 at 15:48

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.