Cohomological injectivity of a degree-one map

A degree-one map $f:M\to N$ between closed connected oriented manifolds of the same dimension induces an injection $f^{\ast }:H^{\ast }(N;R)\to H^{\ast }(M;R)$. The wrong-way map defined using Poincare duality is a left inverse by the projection formula and $f_{\ast }[M]=[N]$.