Maximal first Betti number drop and collapsing RCD spaces
Abstract
Let (Xi,di,mi) be compact RCD(K,N) spaces converging to a space X of rectifiable dimension m. Their first Betti numbers can drop by at most N−m. When equality holds, the maximal abelian covers have a non-collapsed limit Y carrying a free isometric ℝN−m-action. We turn this limiting symmetry into fibrations of the approximating spaces. More precisely, after passing to a subsequence, there is an open full-measure set G⊂ X, containing every regular point, on which X is a topological orbifold and the Xi admit local Seifert fibrations with (N−m)-torus fibres. Each finite local group acts on the fibre by translations. If the Xi have no boundary, then X∖ G has Hausdorff codimension at least two. If Y has no bubbling and X is a smooth closed Riemannian orbifold, the local fibrations may be chosen as restrictions of a single global Seifert fibration. An affine replacement on the smooth manifold cover shows, in addition, that a finite cover of Xi is homeomorphic to a product with 𝕋N−m; here we use the classification of affine torus bundles at the Betti number equality of Peng, Wang and Wang. When the base is a closed Riemannian manifold, the maps are torus bundles, confirming a conjecture of Zamora and Zhu for possibly singular RCD total spaces. The new ingredients are a pointwise linearization of the collapsing action at regular orbits, an invariant harmonic transverse coordinate compatible with finite isotropy, and an exactly equivariant orbit coordinate built from an elementary rounding-and-doubling argument in the collapsing deck group.
Related
- The first ℓ2-Betti number, cost and rank gradient of few-relator random groups
- Odd-rank maximal ideals at collapsing levels of type D
- A Sharp Bound for the Second Betti Number of Hyperkähler Sixfolds
- The Singular Fibration of Collapsing RCD Spaces with Reifenberg Local Covers
- On leading Gopakumar--Vafa invariants of local ℙ2 and leading Betti numbers of moduli spaces of one-dimensional sheaves on ℙ2
BibTeX
@misc{huang2026maximal,
title = {{Maximal first Betti number drop and collapsing RCD spaces}},
author = {Shaosai Huang and Xin Peng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30153},
eprint = {2609.30153},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30153},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Maximal first Betti number drop and collapsing RCD spaces
AU - Shaosai Huang
AU - Xin Peng
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30153
UR - https://arxiv.org/abs/2609.30153
AB - Let (X_i,d_i,m_i) be compact RCD(K,N) spaces converging to a space X of rectifiable dimension m. Their first Betti numbers can drop by at most N−m. When equality holds, the maximal abelian covers have a non-collapsed limit Y carrying a free isometric ℝ^N−m-action. We turn this limiting symmetry into fibrations of the approximating spaces. More precisely, after passing to a subsequence, there is an open full-measure set G⊂ X, containing every regular point, on which X is a topological orbifold and the X_i admit local Seifert fibrations with (N−m)-torus fibres. Each finite local group acts on the fibre by translations. If the X_i have no boundary, then X∖ G has Hausdorff codimension at least two. If Y has no bubbling and X is a smooth closed Riemannian orbifold, the local fibrations may be chosen as restrictions of a single global Seifert fibration. An affine replacement on the smooth manifold cover shows, in addition, that a finite cover of X_i is homeomorphic to a product with 𝕋^N−m; here we use the classification of affine torus bundles at the Betti number equality of Peng, Wang and Wang. When the base is a closed Riemannian manifold, the maps are torus bundles, confirming a conjecture of Zamora and Zhu for possibly singular RCD total spaces. The new ingredients are a pointwise linearization of the collapsing action at regular orbits, an invariant harmonic transverse coordinate compatible with finite isotropy, and an exactly equivariant orbit coordinate built from an elementary rounding-and-doubling argument in the collapsing deck group.
ER -