TITLE: Are there two non-diffeomorphic smooth manifolds with the same homology groups?
QUESTION [6 upvotes]: I know that there definitely are two topological spaces with the same homology groups, but which are not homeomorphic. For example, one could take $T^{2}$ and $S^{1}\vee S^{1}\vee S^{2}$ (or maybe $S^{1}\wedge S^{1}\wedge S^{2}$), which have the same homology groups but different fundamental groups. But are there any examples in the smooth category?

REPLY [5 votes]: Serre has shown with the help of two embeddings phi and psi of a quadratic number field into C that there exist two projective surfaces V(phi)and V(psi) over C which have non isomorphic fundamental groups (and so are non homeomorphic) but have isomorphic Betti numbers.
The comparison theorem between étale cohomology and singular cohomology ( which didn't exist when Serre wrote his article )  even proves thar these surfaces have the same singular cohomology with value in any finite abelian group or over Q_l(l-adics) for any prime l.
I don't know if these surfaces have the same homology and so I don't answer your question in the strict sense (anyway, now you have Andy's and Eric's most satisfying solutions); but these remarks in an algebraic geometry context might interest you.
Serre's article is 
Exemples de variétés projectives conjuguées non homéomorphes,  C.R. Acad.Sci.Paris 258 (1964), 4194-4196
It is of course reproduced in his Collected Papers.