As a consequence of the Whitehead theorem, Spanier's Algebraic Topology book has on 7.6.25 the following theorem:
A weak homotopy equivalence induces isomorphisms of the corresponding integral singular homology. Conversely, a map between simply connected spaces which induces isomorphisms of the corresponding integral singular homology groups is a weak homotopy equivalence.
Is it possible that a map between (non-simply connected) topological spaces induces an isomorphism on all homology groups, and yet is not a weak homotopy equivalence? If so, I would be glad to have an example.