Singular sets on spaces with integral curvature bounds and diffeomorphism finiteness for manifolds

In this paper, we are concerned with noncollapsed Riemannian manifolds $(M^n,g)$ with integral curvature bounds, as well as their Gromov-Hausdorff limits $(M_i^n,g_i)\stackrel{GH}{\to }(X,d)$. Our main result generalizes Cheeger’s Hausdorff dimension estimate for the singular set in [7] and improve it into Minkowski dimension estimate in the spirit of Cheeger-Naber [12]. We also prove a difffeomorphism finiteness theorem for manifolds in the critical case, using Cheeger-Naber’s methods in [13].