Lower Ricci curvature and nonexistence of manifold structure

We build limit spaces .M n j ; g j / !.X k ; d / of manifolds M j with uniform lower bounds on Ricci curvature such that X k is nowhere a topological manifold, and in fact every open set U X has infinitely generated homology.More completely, it is known that any such X k must be k-rectifiable for some unique dim X WD k n D dim M j .It is also known that if k D 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 Ric > and 2 R. Then for each > 0 we construct a complete 4-rectifiable metric space .X 4 ; d / with d GH .X 4 ; X 4 / < such that the following hold.First, X 4 is a limit space .M 6 j ; g j / !X 4 , where M 6 j are smooth manifolds satisfying the same lower Ricci bound Ric j > .Additionally, X 4 has no open subset which is topologically a manifold.Indeed, for any open U X 4 we have that the second homology H 2 .U / is infinitely generated.Topologically, X 4 is the connect sum of X 4 with an infinite number of densely spaced copies of CP 2 .In this way we see that every 4-manifold X 4 may be approximated arbitrarily closely by 4-dimensional limit spaces X 4 which are nowhere manifolds.We will see that there is a sense, as yet imprecise, in which generically one should expect manifold structures to not exist on spaces with higher-dimensional Ricci curvature lower bounds.