Fixed points of $S^2\times S^2$ and $S^2\times S^2\times S^2$.

Question

Consider the rotation about the $z$-axis on $S^2$. We know that $S^2$ has two fixed points which are north and South Pole.

I want to learn the fixed points of $S^2\times S^2$ and of $S^2\times S^2\times S^2$ with respect to rotation.

Answer 1

It depends if you rotate one of the factors or all of them.

In you rotated just the first factor then you obtain two copies of $S^{2}$ or $S^{2} \times S^{2}$ respectively.

If you rotate all the factors then you obtain $4$ or $8$ isolated fixed points respectively.