TITLE: A map inducing isomorphisms on homology but not on homotopy
QUESTION [10 upvotes]: 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.

REPLY [3 votes]: Several decades ago this question led to the concept of nilpotent space, nilpotent CW complex.  You can google that, along with "Emmanuel Dror", who later changed his name to Emmanuel Dror Farjoun.