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Let $X$ be a hyperbolic 5-manifold. Can there be any class in $H^2(X;\mathbf{Z})$ that is torsion and whose square in $H^4(X;\mathbf{Z})$ is not zero?

For example, are there hyperbolic 5-manifolds that have the same integer cohomology ring as $\mathbf{R}P^5$, or another 5d lens space?

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    $\begingroup$ You probably meant to assume that $X$ is closed or finite volume. In the arithmetic case a lot is known (even though I doubt that complete cohomology ring is ever computed). I would start from the seminal paper [Millson, John J. On the first Betti number of a constant negatively curved manifold. Ann. of Math. (1976)] and use MathSciNet to trace references to the modern times. $\endgroup$
    – Igor Belegradek
    Commented Oct 12, 2016 at 12:32

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