{
  "abstract": "Let $(X_i,d_i,\\mathfrak m_i)$ be compact $\\mathrm{RCD}(K,N)$ 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 $\\mathbb{R}^{N-m}$-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 $G\\subset X$, 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 $X\\setminus G$ 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 $\\mathbb{T}^{N-m}$; 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30153",
  "author_total": 2,
  "authors": [
    {
      "name": "Shaosai Huang"
    },
    {
      "name": "Xin Peng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0c96022z9yefm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30153",
  "fulltext_url": null,
  "id": "0c96022z9yefm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30153",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7214378492",
  "pdf_url": "https://arxiv.org/pdf/2609.30153",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0c96022z9yefm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.30153",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30153"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Maximal first Betti number drop and collapsing RCD spaces",
  "title_text": "Maximal first Betti number drop and collapsing RCD spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/0c96022z9yefm",
  "venue": "arXiv",
  "versions": []
}