A map inducing isomorphisms on homology but not on homotopy 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.
 A: 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.
A: How about a knot complement? It's known that the inclusion of a meridian $S^1$  (a small circle which links the knot exactly once) into the knot
complement $S^3 \setminus S^1$ is a homology isomorphism. It is only an equivalence when the knot is trivial. Of course, this case is detected by the fundamental group.
If you want an example which is not detected by $\pi_1$, it turns out there are non-trivial  high dimensional smooth knots $S^n \subset S^{n+2}$ such that $\pi_1$ of the knot complement $X$ is infinite cyclic. The inclusion of the meridian $S^1 \subset X$ is a homology isomorphism, a $\pi_1$ isomorphism but is never a weak equivalence since by a result of Levine if it were then the knot would be trivial (we should assume $n \ge 3$ here).
A: Let $M$ be the Poincare sphere. It is a 3-manifold which has the homology of $S^3$, but non-trivial fundamental group (the binary icosahedral group). In particular, if we remove a point from $M$ we obtain a space $X$ which has the homology of a point (one can verify this by Mayer--Vietoris) but non-trivial fundamental group.
A: 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. 
