Topology and ϵ–regularity theorems on collapsed manifolds with Ricci curvature bounds

In this paper we discuss and prove ϵ –regularity theorems for Einstein manifolds (Mⁿ, g), and more generally manifolds with just bounded Ricci curvature, in the collapsed setting. A key tool in the regularity theory of noncollapsed Einstein manifolds is the following. If x ϵ Mⁿ is such that Vol(B₁(x)) > υ > 0 and that B₂(x) is sufficiently Gromov– Hausdorff close to a cone space B₂(0^n-ℓ, y*) ⊂ ℝ^n-ℓ ×C(Y^ℓ-1) for ℓ≤ 3, then in fact |Rm| ≤ 1 on B₁(x). No such results are known in the collapsed setting, and in fact it is easy to see that without further assumptions such results are false. It turns out that the failure of such an estimate is related to topology. Our main theorem is that for the above setting in the collapsed context, either the curvature is bounded, or there are topological constraints on B₁(x). More precisely, using established techniques one can see there exists ϵ(n) such that if (Mⁿ; g) is an Einstein manifold and B₂(x) is ϵ–Gromov–Hausdorff close to ball in B₂(0^{k- ℓ}, z*) ⊂ℝ^k-ℓ × Z^ℓ, then the fibered fundamental group Γ_ϵ(x) ≡ Image[π₁(B_ϵ(x))→π₁(B₂(x))is almost nilpotent with rank. Γ_ϵ(x)) ≤ n-k. The main result of the this paper states that if rank(Γ_ϵ(x)) = n - k is maximal, then |Rm| ≤ C on B₁(x). In the case when the ball is close to Euclidean, this is both a necessary and sufficient condition. There are generalizations of this result to bounded Ricci curvature and even just lower Ricci curvature.