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I just complemented the question with the missing mathjax syntax, so that it would be more readable.
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edit approved Jan 11, 2021 at 14:16
Duarte Costa
edit approved Jan 11, 2021 at 14:16
Duarte Costa
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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$T^{2}$ and S^{1}\vee S^{1}\vee S^{2} $S^{1}\vee S^{1}\vee S^{2}$ (or maybe S^{1}\wedge S^{1}\wedge S^{2}$S^{1}\wedge S^{1}\wedge S^{2}$), which have the same homology groups but different fundamental groups. But are there any examples in the smooth category?

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?

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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Kirill Levin
Kirill Levin
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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?