Topological singularities in the codimension-two stratum of noncollapsed Ricci limits

Nifa, Anass

doi:10.5281/zenodo.23049912

Preprint · v2

Topological singularities in the codimension-two stratum of noncollapsed Ricci limits

Anass Nifa31 July 2026

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Abstract

We construct classical noncollapsed Ricci limits with a topologically singular point whose unique tangent cone has a topologically Euclidean vertex. Fix an integer $n\ge4$ and a closed connected oriented smooth manifold $Q^n$ such that, for some smoothly embedded closed $n$ -disc $D^n\subset Q$ , $Q\setminus\operatorname{int}D^n$ carries a positive-Ricci metric with unit-round strictly convex boundary. For all sufficiently small $q>0$ , there is a complete pointed $(n+1)$ -dimensional Gromov–Hausdorff limit $(Y, d, p)$ , realized by complete smooth boundaryless $(n+1)$ -manifolds with nonnegative Ricci curvature and uniformly positive based unit-ball volumes, with unique tangent cone $\mathbb{R}^{n-1}\times\operatorname{Cone}(S^1_{2\pi q})$ at $p$ , where $\operatorname{Cone}(S^1_{2\pi q})$ is the cone over the circle of length $2\pi q$ . A neighbourhood of $p$ is homeomorphic to the open topological cone over $Q\#(-Q)$ , where $-Q$ has opposite orientation, and $H_j(Y, Y\setminus\{p\};A)\cong\widetilde H_{j-1}(Q\#(-Q);A)$ for every abelian group $A$ and $j\ge0$ . If $Q\#(-Q)$ is not an integral homology $n$ -sphere, then $p$ is topologically singular. At $p$ , the density is $q$ , and $Y$ has empty Kapovitch–Mondino boundary. For each integer $N\ge5$ , there is an $N$ -dimensional example where $p$ is the unique nonmanifold point and every tangent cone at every point splits $\mathbb{R}^{N-2}$ . Thus the Cheeger–Colding codimension-four singular stratum is empty although $Y$ is not a topological manifold, disproving the Cheeger–Colding conjecture that the interior of its complement is a topological manifold.

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
57P05Secondary: Local properties of generalized manifolds
55N10Secondary: Singular homology and cohomology theory

Mathematics Subject Classification 2020

Record

Version DOI: 10.5281/zenodo.23049912
All versions: 10.5281/zenodo.22670410
Subjects: Ricci curvature; Topological singularities
Research index: OpenAlex record
Language: English

Cite this paper

Anass Nifa. Topological singularities in the codimension-two stratum of noncollapsed Ricci limits. Preprint, Zenodo, v2 (2026). doi:10.5281/zenodo.23049912

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References

As listed in the manuscript bibliography.

[1]
M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren der mathematischen Wissenschaften, vol. 319, Springer, Berlin, 1999.
[2]
D. Burago, Yu. Burago, and S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001.
[3]
B. L. Burdick, Ricci-positive metrics on connected sums of projective spaces, Differential Geom. Appl. 62 (2019), 212–233.
[4]
E. Bruè, A. Pigati, and D. Semola, Topological regularity and stability of noncollapsed spaces with Ricci curvature bounded below, Duke Math. J., to appear; arXiv:2405.03839.
arXiv
[5]
J. Cheeger and T. H. Colding, On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom. 46 (1997), 406–480.
[6]
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), 407–538.
[7]
J. Cheeger and A. Naber, Regularity of Einstein manifolds and the codimension 4 conjecture, Ann. of Math. (2) 182 (2015), 1093–1165.
[8]
T. H. Colding and A. Naber, Characterization of tangent cones of noncollapsed limits with lower Ricci bounds and applications, Geom. Funct. Anal. 23 (2013), 134–148.
[9]
G. De Philippis and N. Gigli, Non-collapsed spaces with Ricci curvature bounded from below, J. Éc. polytech. Math. 5 (2018), 613–650.
[10]
V. Kapovitch and A. Mondino, On the topology and the boundary of 𝑁-dimensional RCD(𝐾, 𝑁) spaces, Geom. Topol. 25 (2021), 445–495.
[11]
X. Menguy, Noncollapsing examples with positive Ricci curvature and infinite topological type, Geom. Funct. Anal. 10 (2000), no. 3, 600–627.
[12]
J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286–294.
[13]
P. Reiser, Examples of tangent cones of non-collapsed Ricci limit spaces, Nonlinear Anal. 252 (2025), Article 113699.
[14]
J. Milnor, Morse Theory, Annals of Mathematics Studies, vol. 51, Princeton University Press, Princeton, 1963.
[15]
P. Reiser, Positive Ricci curvature on connected sums of fibre bundles, arXiv:2406.02274.
arXiv
[16]
P. Reiser and D. J. Wraith, A generalization of the Perelman gluing theorem and applications, arXiv:2308.06996v2.
arXiv v2
[17]
K.-T. Sturm, On the geometry of metric measure spaces. II, Acta Math. 196 (2006), 133–177.

[reference-1] [1]
M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren der mathematischen Wissenschaften, vol. 319, Springer, Berlin, 1999.

[reference-2] [2]
D. Burago, Yu. Burago, and S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001.

[reference-3] [3]
B. L. Burdick, Ricci-positive metrics on connected sums of projective spaces, Differential Geom. Appl. 62 (2019), 212–233.

[reference-4] [4]
E. Bruè, A. Pigati, and D. Semola, Topological regularity and stability of noncollapsed spaces with Ricci curvature bounded below, Duke Math. J., to appear; arXiv:2405.03839.
arXiv

[reference-5] [5]
J. Cheeger and T. H. Colding, On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom. 46 (1997), 406–480.

[reference-6] [6]
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), 407–538.

[reference-7] [7]
J. Cheeger and A. Naber, Regularity of Einstein manifolds and the codimension 4 conjecture, Ann. of Math. (2) 182 (2015), 1093–1165.

[reference-8] [8]
T. H. Colding and A. Naber, Characterization of tangent cones of noncollapsed limits with lower Ricci bounds and applications, Geom. Funct. Anal. 23 (2013), 134–148.

[reference-9] [9]
G. De Philippis and N. Gigli, Non-collapsed spaces with Ricci curvature bounded from below, J. Éc. polytech. Math. 5 (2018), 613–650.

[reference-10] [10]
V. Kapovitch and A. Mondino, On the topology and the boundary of 𝑁-dimensional RCD(𝐾, 𝑁) spaces, Geom. Topol. 25 (2021), 445–495.

[reference-11] [11]
X. Menguy, Noncollapsing examples with positive Ricci curvature and infinite topological type, Geom. Funct. Anal. 10 (2000), no. 3, 600–627.

[reference-12] [12]
J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286–294.

[reference-13] [13]
P. Reiser, Examples of tangent cones of non-collapsed Ricci limit spaces, Nonlinear Anal. 252 (2025), Article 113699.

[reference-14] [14]
J. Milnor, Morse Theory, Annals of Mathematics Studies, vol. 51, Princeton University Press, Princeton, 1963.

[reference-15] [15]
P. Reiser, Positive Ricci curvature on connected sums of fibre bundles, arXiv:2406.02274.
arXiv

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

[reference-17] [17]
K.-T. Sturm, On the geometry of metric measure spaces. II, Acta Math. 196 (2006), 133–177.