Suppose I have topological spaces $X, Y$ and a continuous map $f: X \to Y$. Let $\mathbb{k}$ be a field, and $i \ge 1$ an integer.
If the induced linear map on homology $f_* : H_i ( X, \mathbb{k}) \to H_i ( Y, \mathbb{k}) $ is surjective, does it necessarily follow that the induced map
linear map on cohomology $f^* : H^i ( Y, \mathbb{k}) \to H^i ( X, \mathbb{k}) $ is injective?
(In case it makes any difference: in the case I am interested in, $X$ and $Y$ are finite simplicial complexes, $X$ is a subcomplex of $Y$, and $f$ is the inclusion map.)
Best Answer
Hints: If $F$ is a field then it follows from the universal coefficient theorem (for homology and co-homology) that $H^*(X;F)= Hom_{F-mods}(H_*(X;F);F)$.
This implies if $f_*$ is surjective then $f^*$ is injective.