Given a map between two manifolds that induces an isomorphism on integral cohomology in the top dimension, it follows from naturality of the cup product and Poincaré duality and universal coefficient theorem that all maps on cohomology in every dimension are injective with torsion free cokernel.


Is there an example where one of these maps is not surjective?



map of manifolds inducing iso on top cohomology, but not surjective on one other cohomology group
For any orientable manifold M of dimension n, map M to $S^n$ by sending some suitable neighborhood of a point homeomorphically to $\mathbb{R}^n$ and the rest to $\infty$.


