I'm looking for an example of a continuous map $f:X\to Y$ between topological spaces that induces isomorphisms on all homology groups $f_\ast:H_n(X)\overset{\cong}{\longrightarrow} H_n(Y)\ \forall n\geq 0$, i.e. a quasi-isomorphism, such that the induced morphism on homotopy groups $f_\ast:\pi_n(X)\to\pi_n(Y)$ is not an isomorphism for all $n$, i.e. $f$ is not a weak homotopy equivalence.
I've tried to come up with examples by taking two non homotopy equivalent CW complexes with isomorphic homology groups, such as $S^1\times S^1$ and $S^1\vee S^1\vee S^2$, and finding such an $f$ for that case as then Whitehead's theorem would imply $f$ is not a weak homotopy equivalence. The problem is finding the map, since just having isomorphic homology groups is not enough, we need the isomorphisms to be induced by a single map.
Another idea was to look for a space with all homology groups being trivial but with some non-trivial homotopy groups, as then a constant map (sending everything to a single point) would suffice as an example. But this idea was also unsuccesful as we have to take into account Hurewicz theorem while constructing such a space...
Does anybody know of an example of such a map? Moreover, is there an example where the map is between CW complexes?
Thanks in advance.
