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Let $ p$ be an odd prime. Can we construct a free action of the cyclic group $\Bbb Z/p\Bbb Z$ on $S^n \times \cdots \times S^n$($n$ is odd), which is not conjugate to the free action given by multiplying some coordinate by $e^{2\pi i/p} $ or $e^{-2\pi i/p}$? I have tried to construct such examples but could not find any.

Thank you so much in advance.

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    $\begingroup$ You can conjugate multiplication by $e^{-2\pi i/3}$ by any automorphism of the space. Do you perhaps want to consider an action which is not conjugate to this action? $\endgroup$
    – Wojowu
    Commented Jan 6, 2019 at 18:45
  • $\begingroup$ @Wojowu Yes, I want to consider an action which is no conjugate to this action. $\endgroup$
    – student
    Commented Jan 6, 2019 at 19:52
  • $\begingroup$ @Wojowu Also, I would like to see some examples of conjugate multiplication by $e^{−2πi/3}$ by any automorphism of the space which is not exactly same as multiplication one coordinate by $e^{2πi/3}$ or $e^{−2πi/3}$. Thanks. $\endgroup$
    – student
    Commented Jan 6, 2019 at 20:23

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For one factor of $S^n$, it seems that what you are asking about would be referred to as a "fake lens space". (More properly, the quotient would be a fake lens space, and the action is the action on the universal covering space.) For $n\geq 5$, these are classified in chapter 14E of Wall's book, Surgery on Compact Manifolds; the answer is a bit easier to understand when $p$ is odd. The methods are those of surgery theory, so it's not so easy to describe the action of the generator of $\mathbb{Z}/p\mathbb{Z}$.

You can apply similar methods to products of spheres.

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