Cohomology ring of a finite cyclic group over its prime field ($H^{\ast }(B{C}_p;{\mathbb{F}}_p)$)

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, andOne 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 $H^{\ast }(B{C}_2;{\mathbb{F}}_2)={\mathbb{F}}_2[v]$ 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.