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In this paper we discuss and prove $\epsilon$-regularity theorems forEinstein manifolds $(M^n,g)$, and more generally manifolds with just boundedRicci curvature, in the collapsed setting.  A key tool in the regularity theory of noncollapsed Einstein manifolds is thefollowing: If $x\in M^n$ is such that $Vol(B_1(x))>v>0$ and that $B_2(x)$ issufficiently Gromov-Hausdorff close to a cone space $B_2(0^{n-\ell},y^*)\subset\mathbb{R}^{n-\ell}\times C(Y^{\ell-1})$ for $\ell\leq 3$, then in fact$|Rm|\leq 1$ on $B_1(x)$. No such results are known in the collapsed setting,and in fact it is easy to see without more such results are false. It turns outthat the failure of such an estimate is related to topology. Our main theoremis that for the above setting in the collapsed context, either the curvature isbounded, or there are topological constraints on $B_1(x)$.  More precisely, using established techniques one can see there exists$\epsilon(n)$ such that if $(M^n,g)$ is an Einstein manifold and $B_2(x)$ is$\epsilon$-Gromov-Hausdorff close to ball in$B_2(0^{k-\ell},z^*)\subset\mathbb{R}^{k-\ell}\times Z^\ell$, then the fiberedfundamental group $\Gamma_\epsilon(x)\equivImage[\pi_1(B_\epsilon(x))\to\pi_1(B_2(x))]$ is almost nilpotent with$rank(\Gamma_{\epsilon}(x))\leq n-k$. The main result of the this paper statesthat if $rank(\Gamma_{\epsilon}(x))= n-k$ is maximal, then $|Rm|\leq C$ on$B_1(x)$. In the case when the ball is close to Euclidean, this is both anecessary and sufficient condition. There are generalizations of this result tobounded Ricci curvature and even just lower Ricci curvature.

Topology and \epsilon-regularity Theorems on Collapsed Manifolds with Ricci Curvature Bounds