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let $S$ be the d-sphere. we know that $S \wedge X = \Sigma^d X$ the $d$-fold suspension of $X$. what can we say about $(S\times S) \wedge (S\times S)$ in terms of suspension?

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  • $\begingroup$ This question is more appropriate for math.stackexchange.com $\endgroup$
    – Sean Tilson
    Commented Apr 7, 2011 at 0:02

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That space it not a suspension. For instance, it has nontrivial products in its cohomology, unlike suspensions. If you suspend your space once, it falls apart as a wedge of various spheres of dimensions d and 2d.

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