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Since it's fairly easy to come up with a two spaces that have different homotopy groups but the same homology groups ($S^2\times S^4$ and $\mathbb{C}\textrm{P}^3$). Are there any nice examples of spaces going the other way around? Are there any obvious ways to approach a problem like this?

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Related: – Grigory M Jan 15 '12 at 16:39
It's worth noting that $S^2\times S^4$ and $\mathbb{CP}^3$ have the same cohomology groups as well, but they have different cohomology rings. – Michael Albanese Jun 21 at 23:16

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up vote 6 down vote accepted

Standard example is $\mathbb RP^2\times S^3$ and $\mathbb RP^3\times S^2$ (they have same homotopy groups since they both have $\pi_1=\mathbb Z/2$ and the universal cover is in both cases $S^2\times S^3$).

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