# Are there two non-diffeomorphic smooth manifolds with the same homology groups?

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?

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The Lens spaces $L(p,q_1)$ and $L(p,q_2)$ for suitable choices of $q_i$'s are homotopy equivalent but not homeomorphic. For example, one can take $L(7,1)$ and $L(7,2)$. –  Somnath Basu Apr 27 '10 at 0:12

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.

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More surprisingly, you can find smooth manifolds which are homeomorphic (and in particular, have the same homology) but are not diffeomorphic! The best-known examples are exotic spheres.

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I actually knew that. For some reason, I didn't make the connection between homeomorphism and same homology groups. Thanks! –  Kirill Levin Oct 30 '09 at 4:11

A more trivial example is R^n and R^m for m and n different. (More generally two contractible spaces of different dimensions.)

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I was thinking of a circle and an annulus, you beat me :-) –  Patrick I-Z Jan 6 '11 at 0:29