Finite Diffeomorphism Theorem for manifolds with lower Ricci curvature and bounded energy

In this paper we prove that the space $\text{cM}(n,\text{rv},D,\mathrm{\Lambda }):=\{(M^n,g)\text{ closed }:\text{Ric}\ge -(n-1),\text{Vol}(M)\ge \text{rv}>0,\text{diam}(M)\le D\text{ and }{\int }_M\mathrm{\mid }\text{Rm}{\mathrm{\mid }}^{n\mathrm{/}2}\le \mathrm{\Lambda }\}$ has at most $C(n,\text{rv},D,\mathrm{\Lambda })$ many diffeomorphism types. This removes the upper Ricci curvature bound of Anderson-Cheeger’s finite diffeomorphism theorem in \cite{AnCh}. Furthermore, if $M$ is Kähler surface, the Riemann curvature $L^2$ bound could be replaced by the scalar curvature $L^2$ bound.