  In this paper we discuss and prove $\epsilon$-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 $x\in M^n$ is such that $Vol(B_1(x))>v>0$ and that $B_2(x)$ is
sufficiently Gromov-Hausdorff close to a cone space $B_2(0^{n-\ell},y^*)\subset
\mathbb{R}^{n-\ell}\times C(Y^{\ell-1})$ 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 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
$\epsilon(n)$ 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 fibered
fundamental group $\Gamma_\epsilon(x)\equiv
Image[\pi_1(B_\epsilon(x))\to\pi_1(B_2(x))]$ is almost nilpotent with
$rank(\Gamma_{\epsilon}(x))\leq n-k$. The main result of the this paper states
that 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 a
necessary and sufficient condition. There are generalizations of this result to
bounded Ricci curvature and even just lower Ricci curvature.
