= Rational cohomology injectivity of a nonzero-degree map
For a <continuous map> $f:M\to N$ of nonzero <mapping degree> between closed connected oriented manifolds of the same dimension, $f^*:H^q(N;\mathbb Q)\to H^q(M;\mathbb Q)$ is injective in every degree. Pair a nonzero class $a$ with a complementary class $b$ using <Poincare duality>. Naturality gives $\langle f^*a\smile f^*b,[M]\rangle=(\deg f)\langle a\smile b,[N]\rangle\ne0$, so $f^*a\ne0$. Over coefficients of positive characteristic the argument requires that the degree remain nonzero in that field.
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