I am looking for references on homology complex projective spaces; or more precisely the classification (if any) of smooth oriented manifolds which have the same homology groups as $\mathbb{CP}^n$.
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4$\begingroup$ The connected sum of $\mathbb{CP}^n$ with any homology sphere will be such a manifold, but there could be others. I don't know what kind of classification you can hope for. $\endgroup$– Michael AlbaneseMay 22, 2022 at 21:50
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1$\begingroup$ It will contain at least all these: map.mpim-bonn.mpg.de/Fake_complex_projective_spaces $\endgroup$– Drew HeardMay 23, 2022 at 6:28
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