The Singular Fibration of Collapsing RCD Spaces with Reifenberg Local Covers
Abstract
Let compact RCD(κ,N) spaces converge in the measured Gromov--Hausdorff sense, and suppose that they have (δ,ρ)-Reifenberg local covers for one fixed ρ>0, with |κ|ρ2≤1 and δ small. We prove that, along a subsequence, the spaces are singular fibrations over the limit: their canonical frame spaces converge to a topological manifold with a locally linear action of O(n) whose quotient is the limit; for large indices the frame spaces fibre over this manifold, by maps that commute with O(n), with fibres homeomorphic to nilmanifolds; and the induced maps to the limit are, near every point, quotients of products by the isotropy group, with fibres homeomorphic to infranilmanifolds. As applications, we extend a fibration theorem of Honda and Zhang to non-smooth RCD spaces whose local covers have injectivity radius bounded below, and we show that for aspherical spaces the isotropy groups are finite, the fibre groups inject into the fundamental groups, and the limit is the quotient of a contractible manifold by a proper cocompact action; under Rong's condition As it is a closed aspherical manifold. Three appendices treat smooth manifolds with Ric≥−(n−1) and small local rewinding Reifenberg error: a uniform short-time Ricci flow with |Rm|≤α/t, also for complete non-compact data, where it is unique among complete flows with curvature at most A/t; a second construction with affine nilmanifold bundles; and, conditional on Nifa's construction, examples showing that rewinding volume control alone does not give continuous Gromov--Hausdorff approximations with small error.
Related
- Xiao's Genus-Two Fibration: Branched Covers and Braid Monodromy
- Parabolic Reifenberg Parametrizations and Corona Decompositions, with Applications to Singular Integrals
- Holomorphic Tischler fibration theorem for Stein spaces
- Maximal first Betti number drop and collapsing RCD spaces
- On finite covers of projective spaces, generic outside of codimension three
BibTeX
@misc{huang2026singular,
title = {{The Singular Fibration of Collapsing RCD Spaces with Reifenberg Local Covers}},
author = {Shaosai Huang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17229},
eprint = {2610.17229},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17229},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - The Singular Fibration of Collapsing RCD Spaces with Reifenberg Local Covers
AU - Shaosai Huang
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.17229
UR - https://arxiv.org/abs/2610.17229
AB - Let compact RCD(κ,N) spaces converge in the measured Gromov--Hausdorff sense, and suppose that they have (δ,ρ)-Reifenberg local covers for one fixed ρ>0, with |κ|ρ^2≤1 and δ small. We prove that, along a subsequence, the spaces are singular fibrations over the limit: their canonical frame spaces converge to a topological manifold with a locally linear action of O(n) whose quotient is the limit; for large indices the frame spaces fibre over this manifold, by maps that commute with O(n), with fibres homeomorphic to nilmanifolds; and the induced maps to the limit are, near every point, quotients of products by the isotropy group, with fibres homeomorphic to infranilmanifolds. As applications, we extend a fibration theorem of Honda and Zhang to non-smooth RCD spaces whose local covers have injectivity radius bounded below, and we show that for aspherical spaces the isotropy groups are finite, the fibre groups inject into the fundamental groups, and the limit is the quotient of a contractible manifold by a proper cocompact action; under Rong's condition A_s it is a closed aspherical manifold. Three appendices treat smooth manifolds with Ric≥−(n−1) and small local rewinding Reifenberg error: a uniform short-time Ricci flow with |Rm|≤α/t, also for complete non-compact data, where it is unique among complete flows with curvature at most A/t; a second construction with affine nilmanifold bundles; and, conditional on Nifa's construction, examples showing that rewinding volume control alone does not give continuous Gromov--Hausdorff approximations with small error.
ER -