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 $f^{\ast }:H^{\ast }(N;F)\to H^{\ast }(M;F)$ is injective. For a nonzero class $a$, Poincare duality supplies $b$ with $⟨a⌣b,[N]⟩\ne 0$, and naturality gives