Homological obstructions to local contractibility in noncollapsed Ricci limits

Nifa, Anass

doi:10.5281/zenodo.23049905

Preprint · v3

Homological obstructions to local contractibility in noncollapsed Ricci limits

Anass Nifa25 August 2026

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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.

MSC 2020

53C23Primary: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C21Secondary: Methods of global Riemannian geometry, including PDE methods; curvature restrictions
55N10Secondary: Singular homology and cohomology theory

Mathematics Subject Classification 2020

Record

Version DOI: 10.5281/zenodo.23049905
All versions: 10.5281/zenodo.22674067
Subjects: Ricci curvature; Local contractibility
Research index: OpenAlex record
Language: English

Cite this paper

Anass Nifa. Homological obstructions to local contractibility in noncollapsed Ricci limits. Preprint, Zenodo, v3 (2026). doi:10.5281/zenodo.23049905

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References

As listed in the manuscript bibliography.

[1]
D. Burago, Yu. Burago, and S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics 33, American Mathematical Society, Providence, RI, 2001.
[2]
E. Bruè, A. Pigati, and D. Semola, Topological regularity and stability of noncollapsed spaces with Ricci curvature bounded below, Duke Math. J., accepted. Accepted manuscript, 107 pp., CVGMT manuscript; original preprint arXiv:2405.03839.
arXiv · Source
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DOI · arXiv v3
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DOI · arXiv v3
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A. Nifa, Topological singularities in the codimension-two stratum of noncollapsed Ricci limits, author’s preprint, 2026, 94 pp. doi: 10.5281/zenodo.22670411.
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arXiv v1
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arXiv v2

[reference-1] [1]
D. Burago, Yu. Burago, and S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics 33, American Mathematical Society, Providence, RI, 2001.

[reference-2] [2]
E. Bruè, A. Pigati, and D. Semola, Topological regularity and stability of noncollapsed spaces with Ricci curvature bounded below, Duke Math. J., accepted. Accepted manuscript, 107 pp., CVGMT manuscript; original preprint arXiv:2405.03839.
arXiv · Source

[reference-3] [3]
B. L. Burdick, Ricci-positive metrics on connected sums of projective spaces, Differential Geom. Appl. 62 (2019), 212–233. doi: 10.1016/j.difgeo.2018.11.005. Theorem C also appears in arXiv:1705.05055v3, p. 3.
DOI · arXiv v3

[reference-4] [4]
J. Cheeger and T. H. Colding, On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom. 46 (1997), no. 3, 406–480. doi: 10.4310/jdg/1214459974.
DOI

[reference-5] [5]
J. Cheeger, W. Jiang, and A. Naber, Rectifiability of singular sets of noncollapsed limit spaces with Ricci curvature bounded below, Ann. of Math. (2) 193 (2021), no. 2, 407–538. doi: 10.4007/annals.2021.193.2.2.
DOI

[reference-6] [6]
J. Cheeger and A. Naber, Regularity of Einstein manifolds and the codimension 4 conjecture, Ann. of Math. (2) 182 (2015), no. 3, 1093–1165. doi: 10.4007/annals.2015.182.3.5.
DOI

[reference-7] [7]
T. H. Colding and A. Naber, Characterization of tangent cones of noncollapsed limits with lower Ricci bounds and applications, Geom. Funct. Anal. 23 (2013), no. 1, 134–148. doi: 10.1007/s00039-012-0202-7. The lemma and equation numbers refer to arXiv:1108.3244v3.
DOI · arXiv v3

[reference-8] [8]
A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.

[reference-9] [9]
E. Hupp, A. Naber, and K.-H. Wang, Lower Ricci curvature and nonexistence of manifold structure, Geom. Topol. 29 (2025), no. 1, 443–477. doi: 10.2140/gt.2025.29.443. Preprint arXiv:2308.03909v1.
DOI · arXiv v1

[reference-10] [10]
N. Li and A. Naber, Quantitative estimates on the singular sets of Alexandrov spaces, Peking Math. J. 3 (2020), no. 2, 203–234. doi: 10.1007/s42543-020-00026-2. Preprint arXiv:1912.03615v1.
DOI · arXiv v1

[reference-11] [11]
X. Menguy, Noncollapsing examples with positive Ricci curvature and infinite topological type, Geom. Funct. Anal. 10 (2000), 600–627. doi: 10.1007/PL00001632.
DOI

[reference-12] [12]
J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), no. 2, 286–294. doi: 10.1090/S0002-9947-1965-0182927-5.
DOI

[reference-13] [13]
A. Naber, Conjectures and open questions on the structure and regularity of spaces with lower Ricci curvature bounds, SIGMA 16 (2020), Paper No. 104, 8 pp. doi: 10.3842/SIGMA.2020.104.
DOI

[reference-14] [14]
A. Nifa, Topological singularities in the codimension-two stratum of noncollapsed Ricci limits, author’s preprint, 2026, 94 pp. doi: 10.5281/zenodo.22670411.
DOI

[reference-15] [15]
P. Reiser, Positive Ricci curvature on connected sums of fibre bundles, preprint, arXiv:2406.02274v1, 4 June 2024, 33 pp.
arXiv v1

[reference-16] [16]
P. Reiser and D. J. Wraith, A generalization of the Perelman gluing theorem and applications, preprint, arXiv:2308.06996v2, 17 January 2024, 22 pp.
arXiv v2