Lower Ricci Curvature and Nonexistence of Manifold Structure

It is known that a limit $(M^n_j,g_j)\to (X^k,d)$ of manifolds $M_j$ with uniform lower bounds on Ricci curvature must be $k$-rectifiable for some unique $\dim X:= k\leq n = \dim M_j$. It is also known that if $k=n$, then $X^n$ is a topological manifold on an open dense subset, and it has been an open question as to whether this holds for $k<n$. Consider now any smooth complete $4$-manifold $(X^4,h)$ with $\text{Ric}>\lambda$ and $\lambda\in \mathbb{R}$. Then for each $\epsilon>0$ we construct a complete $4$-rectifiable metric space $(X^4_\epsilon,d_\epsilon)$ with $d_{GH}(X^4_\epsilon,X^4)<\epsilon$ such that the following hold. First, $X^4_\epsilon$ is a limit space $(M^6_j,g_j)\to X^4_\epsilon$ where $M^6_j$ are smooth manifolds with $\text{Ric}_j>\lambda$ satisfying the same lower Ricci bound. Additionally, $X^4_\epsilon$ has no open subset which is topologically a manifold. Indeed, for any open $U\subseteq X^4_\epsilon$ we have that the second homology $H_2(U)$ is infinitely generated. Topologically, $X^4_\epsilon$ is the connect sum of $X^4$ with an infinite number of densely spaced copies of $\mathbb{C} P^2$ . In this way we see that every $4$-manifold $X^4$ may be approximated arbitrarily closely by $4$-dimensional limit spaces $X^4_\epsilon$ which are nowhere manifolds. We will see there is an, as now imprecise, sense in which generically one should expect manifold structures to not exist on spaces with higher dimensional Ricci curvature lower bounds.