P-th powers in mod p cohomology of fibers

Abstract

Let $F\hookrightarrow E\twoheadrightarrow B$ be a fibration whose base space $B$ is a finite simply-connected CW-complex of dimension $\le p$ and whose total space $E$ is a path-connected CW-complex of dimension $\le p-1$. If $\alpha \in H^{+}(F;{\mathbb{F}}_p)$ then ${\alpha }^p=0$.

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