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

In this paper, we are concerned with noncollapsed Riemannian manifolds \((M^{n},g)\) $(M^n,g)$ with integral curvature bounds, as well as their Gromov–Hausdorff limits \((M^{n}_{i},g_{i})\xrightarrow {GH}(X,d)\) $(M_i^n,g_i)\stackrel{\mathit{GH}}{\to }(X,d)$ . Our main result generalizes Cheeger’s Hausdorff dimension estimate for the singular set in Cheeger (Geom Funct Anal 13:20–72, 2003) and improve it into Minkowski dimension estimate in the spirit of Cheeger and Naber (Invent Math 191:321–339, 2013). We also prove a diffeomorphism finiteness theorem for manifolds in the critical case, using Cheeger–Naber’s methods in Cheeger and Naber (Ann Math 1093–1165, 2015).