Lower Ricci curvature and nonexistence of manifold structure

Hupp, Erik¹ ; Naber, Aaron² ; Wang, Kai-Hsiang¹

¹ Department of Mathematics, Northwestern University, Evanston, IL, United States
² Institute for Advanced Study, Princeton, NJ, United States

Geometry & topology, Tome 29 (2025) no. 1, pp. 443-477.

Résumé

We build limit spaces $(M_j^n,g_j)\to (X^k,d)$ 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 $U\subseteq X$ has infinitely generated homology.

More completely, it is known that any such $X^k$ must be $k$-rectifiable for some unique $\dim X:=k\le n=\dim M_j$. 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 $\mathrm{Ric}>\lambda$ and $\lambda \in ℝ$. Then for each $𝜖>0$ we construct a complete $4$-rectifiable metric space $(X_{𝜖}^4,d_{𝜖})$ with $d_{GH}(X_{𝜖}^4,X^4)<𝜖$ such that the following hold. First, $X_{𝜖}^4$ is a limit space $(M_j^6,g_j)\to X_{𝜖}^4$, where $M_j^6$ are smooth manifolds satisfying the same lower Ricci bound ${\mathrm{Ric}}_j>\lambda$. Additionally, $X_{𝜖}^4$ has no open subset which is topologically a manifold. Indeed, for any open $U\subseteq X_{𝜖}^4$ we have that the second homology $H_2(U)$ is infinitely generated. Topologically, $X_{𝜖}^4$ 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 $X_{𝜖}^4$ 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.

Détail

DOI : 10.2140/gt.2025.29.443

Keywords: Ricci lower bounds, collapsing, non-manifold structure

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