发表机构
Kspectra Research Inc.; University of Science and Technology of China（Kspectra研究公司; 中国科学技术大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究紧致RCD空间塌缩时第一贝蒂数达到最大下降$N-m$的情形，通过将极限对称性转化为Seifert纤维化，证明有限覆盖与环面乘积同胚，并证实Zamora-Zhu猜想。
AI 中文摘要
设$(X_i,d_i,\mathfrak m_i)$为紧致$\mathrm{RCD}(K,N)$空间，收敛到可修正维数$m$的空间$X$。它们的第一贝蒂数最多下降$N-m$。当等号成立时，最大阿贝尔覆盖具有非塌缩极限$Y$，其上带有自由的等距$\mathbb{R}^{N-m}$作用。我们将此极限对称性转化为逼近空间上的纤维化。更精确地，在取子序列后，存在开的全测度子集$G\subset X$，包含所有正则点，在其上$X$是拓扑orbifold，且$X_i$具有以$(N-m)$维环面为纤维的局部Seifert纤维化。每个有限局部群通过平移作用在纤维上。若$X_i$无边界，则$X\setminus G$的Hausdorff余维数至少为2。若$Y$无气泡且$X$为光滑闭黎曼orbifold，则局部纤维化可选取为单一整体Seifert纤维化的限制。在光滑流形覆盖上的仿射替换进一步表明，$X_i$的有限覆盖与$\mathbb{T}^{N-m}$的乘积同胚；此处我们使用Peng、Wang和Wang在贝蒂数相等情形下的仿射环面丛分类。当基空间为闭黎曼流形时，映射为环面丛，证实了Zamora和Zhu关于可能奇异RCD全空间的猜想。新的工具包括：在正则轨道处塌缩作用的逐点线性化、与有限迷向相容的不变调和横截坐标，以及通过塌缩deck群中初等舍入加倍论证构造的精确等变轨道坐标。
英文摘要
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.
Comments51 pages
