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.
Best Answer
If the isomorphism in homology is meant with integral coefficients (or all constant coefficients), you can take the classifying space $BG$ of any non-trivial discrete acyclic group, for example Higman's four-generator four-relator group (see http://www.encyclopediaofmath.org/index.php/Acyclic_group).
By definition of acyclicity, $H_i(BG;\mathbb{Z})=0$ for $i > 0$ whence (similar as in B. Steinberg's answer) $f: BG \to \ast$ induces an isomorphism on integral homology in all degrees, while $\Pi_1(f): G = \Pi_1(BG) \to \Pi_1(\ast)=0$ is zero.