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

In this paper we discuss and prove is an element of-regularity theorems for Einstein manifolds (M-n, 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 is an element of M-n is such that Vol(B-1 (x)) > v > 0 and that B-2(x) is sufficiently Gromov-Hausdorff close to a cone space B-2(0(n-l) , y*) subset of Rn-l x C(Yl-1) for l <= 3, then in fact vertical bar Rm vertical bar <= 1 on B-1(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 B-1(x) More precisely, using established techniques one can see there exists is an element of(n) such that if (M-n, g) is an Einstein manifold and B-2(x) is is an element of-Gromov-Hausdorff close to ball in B-2(0(k-l), z*) subset of Rk-l x Z(l), then the fibered fundamental group Gamma(is an element of)(x) equivalent to Image [pi(1) (B-is an element of(x)) -> pi(1) (B-2(x))]is almost nilpotent with rank. (Gamma(is an element of)(x)) <= n - k. The main result of the this paper states that if rank (Gamma(is an element of)(x)) <= n - k is maximal, then vertical bar Rm vertical bar <= Con B-1(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.