Lower Ricci Curvature and Nonexistence of Manifold Structure
Erik Hupp, Aaron Naber, Kai-Hsiang Wang
TL;DR
The paper proves that, under a uniform Ricci lower bound, collapsed limit spaces need not admit any open subset that is a topological manifold. It constructs, for any smooth 4-manifold $(X^4,h)$ with $ ext{Ric}_X>\lambda$ and any $oldsymbol{\epsilon}>0$, a nearby 4-rectifiable space $X^4_oldsymbol{\epsilon}$ that arises as a GH limit of smooth 6-manifolds $(M^6_j,g_j)$ with $ ext{Ric}_{g_j}>\lambda$, yet every open set in $X^4_oldsymbol{\epsilon}$ has infinitely generated $H_2$, making $X^4_oldsymbol{\epsilon}$ nowhere a manifold. The construction proceeds by inductively blowing up densely distributed points in $X^4$ to insert $S^2$-factors, controlling Ricci curvature via warped-cone gluings, conical singularities, and bubble-model replacements, and collapsing the added spheres back to points in the limit. This demonstrates that high-dimensional Ricci lower bounds can generically obstruct the existence of manifold structures on limit spaces, highlighting a sharp topological obstruction encoded by infinitely generated second homology. The methods blend warped-product geometry, careful $C^1$ gluing, cone-surgery techniques, and precise control of curvature to realize the desired nonmanifold limit spaces while preserving Ricci bounds in the approximating manifolds.
Abstract
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 $\text{Ric}>λ$ and $λ\in \mathbb{R}$. Then for each $ε>0$ we construct a complete $4$-rectifiable metric space $(X^4_ε,d_ε)$ with $d_{GH}(X^4_ε,X^4)<ε$ such that the following hold. First, $X^4_ε$ is a limit space $(M^6_j,g_j)\to X^4_ε$ where $M^6_j$ are smooth manifolds with $\text{Ric}_j>λ$ satisfying the same lower Ricci bound. Additionally, $X^4_ε$ has no open subset which is topologically a manifold. Indeed, for any open $U\subseteq X^4_ε$ we have that the second homology $H_2(U)$ is infinitely generated. Topologically, $X^4_ε$ 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 $X^4_ε$ 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.