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We consider free actions of $(\mathbb{Z}/p)^2$ on $S^{2n-1}\times S^{2n-1}$given by linear actions of $(\mathbb{Z}/p)^2$ on $\mathbb{R}^{4n}$. Simpleexamples include a lens space cross a lens space, but $k$-invariantcalculations show that other quotients exist. Using the tools of Postnikovtowers and surgery theory, the quotients are classified up to homotopy by the$k$-invariants and up to homeomorphism by the Pontrjagin classes. We willpresent these results and demonstrate how to calculate the $k$-invariants andthe Pontrjagin classes from the rotation numbers.

Linear actions of $\mathbb{Z}/p\times\mathbb{Z}/p$ on $S^{2n-1}\times S^{2n-1}$