Let $F\hookrightarrow E\twoheadrightarrow B$ be a fibration whose base space $B$ is a finite simply-connected CW-complex of dimension $\leq p$ and whose total space $E$ is a path-connected CW-complex of dimension $\leq p-1$. If $\alpha\in H^{+}(F;\mathbb{F}_p)$ then $\alpha ^{p}=0$.
The cohomology of certain groups
On RO(G)-graded equivariant "ordinary" cohomology where G is a power of Z/2
The $ER(2)$-cohomology of $B\mathbb{Z}/(2^q)$ and $\mathbb{C}P^n$
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