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We prove that $b_2$ is bounded for hyperk\"ahler manifolds with vanishingodd-Betti numbers. The explicit upper boundary is conjectured. Following themethod described by Sawon we prove that $b_2$ is bounded in dimension eight andten in the case of vanishing odd-Betti numbers by 24 and 25 respectively.

Boundness of $b_2$ for hyperk\"ahler manifolds with vanishing odd-Betti numbers