I was asking myself the question, if a map $f\colon X \to Y$ between CW complexes gives an isomorphism between $H^*(X,\mathbb{Z})$ and $H^*(Y,\mathbb{Z})$ does it already give an isomorphism between the rational cohomology groups?
The question arose in the context of the following equivalent question: Does a map $f$ between CW complexes, that induces an isomorphism between integral cohomology induce an isomorphism in integral homology.
Note here, that the solution for CW complexes with finitely many cells in each dimension is fairly easy, because then tensoring with the rationals behaves well with the $Hom$-functor or you can use the "doubled" UCT.
Also note, that the equivalence comes from the fact, that cohomology with field coefficients is just the dual of the homology and the fact, that a map is an isomorphism in integral homology iff it induces an isomorphism on homology with finite field coefficients. now you get the fact about the finite field coefficients from the bockstein sequence and the 5-lemma, but the rational part stays a mystery.
