  In this paper we use the Weitzenb\"ock formulas to get information about the
Betti numbers of compact nearly $G_2$ and compact nearly K\"{a}hler
$6$-manifolds. First, we establish estimates on two curvature-type self adjoint
operators on particular spaces assuming bounds on the sectional curvature. Then
using the Weitzenb\"ock formulas on harmonic forms, we get results of the form:
if certain lower bounds hold for these curvature operators then certain Betti
numbers are zero. Finally, we combine both steps above to get sufficient
conditions of vanishing of certain Betti numbers based on the bounds on the
sectional curvature.
