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I wonder if there are examples of 5-dimensional manifolds with vanishing integral second homology group, but non-vanishing second homotopy group? Or is it impossible by some Hurewicz theorem type of argument? Unfortunately I do not see it.

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    $\begingroup$ $\mathbb{R}P^2\times S^3$ appears to be such an example. $\endgroup$
    – Mark Grant
    Commented Sep 13, 2023 at 20:15

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