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In this paper we use the Weitzenb\"ock formulas to get information about theBetti numbers of nearly $G_2$ and nearly K\"ahler manifolds. First, weestablish estimates on two curvature-type self adjoint operators on particularspaces assuming bounds on the sectional curvature. Then using the Weitzenb\"ockformulas on harmonic forms, we get results of the form: if certain lower boundshold for these curvature operators then certain Betti numbers are zero.Finally, we combine both steps above to get sufficient conditions of vanishingof certain Betti numbers based on the bounds on the sectional curvature.

Betti numbers of nearly $G_2$ and nearly K\"ahler manifolds with Weyl curvature bounds