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?
[Math] Are there two non-diffeomorphic smooth manifolds with the same homology groups
homologysmooth-manifolds
Best Answer
Sure -- there are an abundance of homology spheres in dimension 3 (the wikipedia article is pretty nice).
For other examples, in dimension 4 you can find smooth simply-connected closed manifolds whose second homology groups (the only interesting ones) are the same but which have different intersection pairings.
This last subject is very rich. For bathroom reading on it, I cannot recommend Scorpan's book "The Wild World of 4-Manifolds" highly enough.