ℤ∕p× ℤ∕p actions on Sn× Sn

We determine the homotopy type of quotients of S n × S n by free actions of Z /p × Z /p where 2p > n + 3. Much like free Z /p actions, they can be classified via the first p-localized k-invariant, but there are restrictions on the possibilities, and these restrictions are sufficient to determine every possibility in the n = 3 case.We use this to complete the classification of free Z /p × Z /p actions on S 3 × S 3 , for p > 3, by reducing the problem to the simultaneous classification of pairs of binary quadratic forms.Although the restrictions are not sufficient to determine which k-invariants are realizable in general, they can sometimes be used to rule out free actions by groups that contain Z /p × Z /p as a normal Abelian subgroup.