Is it true that if $p: E \to B$ is a $2$-fold covering, the map $p^*$ induced in cohomology is surjective?
2026-03-28 16:17:33.1774714653
Induced map in cohomology of a covering
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If $E=\{1,2\}$, $B=\{3\}$ and $p(x)=3$ for both $x\in E$, then $p^*\colon H^0(B) \to H^0(E)\colon\mathbb{Z}\to\mathbb{Z}^2$ is the map $n\mapsto (n,n)$ which is not surjective.