= Cohomology ring of a finite cyclic group over its prime field
{title2=$H^*(BC_p;\mathbb F_p)$}
The alternating resolution maps $g-1$ and $1+g+\cdots+g^{p-1}$ for a <cyclic group> $C_p$ become zero after applying homomorphisms to its trivial <prime field> module. Thus there is one class in every degree. A two-step shift of the resolution represents a degree-two periodicity class $u$. For odd $p$, a degree-one class $v$ squares to zero by graded commutativity, and
$$
H^*(BC_p;\mathbb F_p)=\Lambda_{\mathbb F_p}(v)\otimes\mathbb F_p[u],\qquad |v|=1,\ |u|=2.
$$
One can take $u=\beta(v)$ for the <Bockstein homomorphism>. For $p=2$ the two maps agree over $\mathbb F_2[C_2]$, so a one-step shift gives $H^*(BC_2;\mathbb F_2)=\mathbb F_2[v]$ with $|v|=1$. The <Künneth theorem> then gives dimension $i+1$ in degree $i$ for $B(C_p\times C_p)$, obstructing <periodic group cohomology> and hence such a free sphere action.
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