Collapsed manifolds with local Ricci bounded covering geometry

For $\rho, v>0$, we say that an $n$-manifold $M$ satisfies local $(\rho,v)$-bound Ricci covering geometry, if Ricci curvature $\text{Ric}_M\ge -(n-1)$, and for all $x\in M$, $\text{vol}(B_\rho(\tilde x))\ge v>0$, where $\tilde x$ is an inverse image of $x$ on the (local) Riemannian universal cover of the $\rho$-ball at $x$. In this paper, we extend the nilpotent fiber bundle theorem of Cheeger-Fukaya-Gromov on a collapsed $n$-manifold $M$ of bounded sectional curvature to $M$ of a local $(\rho,v)$-bound Ricci covering geometry, and $M$ is close to a non-collapsed Riemannian manifold of lower dimension. The nilpotent fiber bundle theorem significantly improves fiber bundle theorem in [Hu], and it strengthens a nilpotent fiber bundle seen from [NZ] and implies the torus bundle in [HW], which are obtained under additional local or global topological conditions, respectively. Our construction of a nilpotent fibration requires a new proof for a result in [HKRX]: if an $n$-manifold $M$ with local $(1,v)$-bound Ricci covering geometry has diameter $<\epsilon(n,v)$, a constant depends on $n$ and $v$, then $M$ is diffeomorphic to an infra-nilmanifold. The proof in [HKRX] is to show that the Ricci flows produces an almost flat metric, thus the result follows from the Gromov's theorem on almost flat manifolds. The new proof is independent of the Gromov's theorem, thus has which as a corollary. If the first Betti number $b_1(M)=n$, then $M$ satisfies a $(1,v)$-bound Ricci covering geometry, thus $M$ is diffeomorphic to a standard torus ([Co2]).