Cohomological injectivity of a map of invertible degree
= Cohomological injectivity of a map of invertible degree
If $f:M\to N$ is a map between closed connected oriented manifolds of the same dimension and its degree is nonzero in a coefficient field $F$, then $f^*:H^*(N;F)\to H^*(M;F)$ is injective. For a nonzero class $a$, <Poincare duality> supplies $b$ with $\langle a\smile b,[N]\rangle\ne0$, and naturality gives
$$
\langle f^*a\smile f^*b,[M]\rangle=(\deg f)\langle a\smile b,[N]\rangle\ne0.
$$