  In this paper, we give an improved Morse index bound of minimal hypersurfaces
from Almgren-Pitts min-max construction in any closed Riemannian manifold
$M^{n+1}$ $(n+1 \geq 3$), which generalizes a result by X. Zhou
\cite{zhou_multiplicity_2019} for $3 \leq n+1 \leq 7$. The novel techniques are
the construction of hierarchical deformations and the restrictive min-max
theory. These techniques do not rely on bumpy metrics, and thus could be
adapted to many other min-max settings.
