Homological obstructions to local contractibility in noncollapsed Ricci limits

Abstract

For every integer $N\ge4$, we construct a compact noncollapsed Ricci limit $X$ of closed $N$-manifolds with nonnegative Ricci curvature such that $0<\mathcal H^{N-2}(S_{\mathrm{top}}(X))<\infty$, where $S_{\mathrm{top}}(X)$ is the nonmanifold locus. There is a compact subset $Z\subset S_{\mathrm{top}}(X)$ of positive $(N-2)$-dimensional Hausdorff measure such that, for every subset $E\subset X$ with $\dim_{\mathcal H}E<N-2$, every relative open neighborhood $U$ in $X\setminus E$ of a point of $Z\setminus E$ satisfies $\operatorname{rank}\operatorname{im}[H_2(U;\mathbb Z)\to H_2(X;\mathbb Z)]=\infty$. Thus no such $E$ has manifold complement, without any closedness or measurability assumption on $E$. These examples attain the codimension-two upper bound for the nonmanifold locus and contradict the Hausdorff-codimension-three manifold conjecture. The realizing sequences have uniformly bounded total scalar curvature and total Ricci norm; in dimensions four and five, their Ricci curvature has a uniform strictly positive lower bound. A separate compact four-dimensional example has exactly one nonmanifold point, with unique tangent cone $\mathbb R^2\times\mathsf C(S^1_{2\pi q})$ for some $q\in(0,1)$, and an empty Cheeger--Colding zero-stratum.