= Free sphere actions exclude elementary abelian subgroups of rank two
{title2=$C_p\times C_p\not\curvearrowright_{\mathrm{free}}S^n\quad(n>1)$}
Suppose a finite <group> $G$ acts freely on $S^n$, $n>1$. The quotient has <fundamental group> $G$ and higher <homotopy groups> equal to those of $S^n$. Attach one $(n+1)$-cell killing a generator of $\pi_n\cong\mathbb Z$, then attach higher cells successively to kill all remaining higher <homotopy>. This constructs a <Eilenberg–MacLane space> $K(G,1)$ with only one $(n+1)$-cell. The <one-cell bound on next-degree cohomology> gives dimension at most one in degree $n+1$. But if $G=C_p\times C_p$, the <mod-p cohomology ring of a cyclic group> and <Künneth theorem> give dimension $n+2$ there. This contradiction excludes that group, and also excludes such a subgroup of any group acting freely on a sphere.
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