Fibrations, the First Betti Number, and Almost Nonnegative Ricci Curvature
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911711197396992 |
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| author | Huang, Hongzhi Huang, Xian-Tao Wang, Jikang Zhu, Xingyu |
| author_facet | Huang, Hongzhi Huang, Xian-Tao Wang, Jikang Zhu, Xingyu |
| contents | In this paper, we prove fibration theorems for manifolds with almost nonnegative Ricci curvature and certain extra regularity assumptions. We show that a closed $n$-manifold $M$ satisfying $\mathrm{diam}(M)^2\mathrm{sec}_M \geq -κ$ and $\mathrm{diam}(M)^2\mathrm{Ric}_M \geq -δ$, where $δ>0$ is sufficiently small depending only on $n$ and $κ$, fibers over a $b_1(M)$-torus. This removes the upper sectional curvature bound required in the earlier result of Yamaguchi \cite{Y88}. As a corollary, we obtain a refinement of Yamaguchi's smooth fibration theorem (\cite{Y91}), showing that the fiber itself (rather than a finite cover of it) fibers over a $b_1$-torus. Our results extend to manifolds satisfying a generalized Reifenberg condition introduced in \cite{HH24}, which encompasses both a lower bound on sectional curvature and the local rewinding Reifenberg condition. In the nonsmooth setting, a similar result also holds for a non-collapsed $\mathrm{RCD}(-ε(D,r,n),n)$ space whose diameter is bounded by $D$ and which satisfies the $(r,δ(n))$-local rewinding Reifenberg condition. The proofs rely on an equivariant regularity theorem for almost submetries under a lower Ricci curvature bound. In addition, we study the stability of rank of Abelian actions along equivariant Gromov-Hausdorff convergence in this paper. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_24380 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fibrations, the First Betti Number, and Almost Nonnegative Ricci Curvature Huang, Hongzhi Huang, Xian-Tao Wang, Jikang Zhu, Xingyu Differential Geometry In this paper, we prove fibration theorems for manifolds with almost nonnegative Ricci curvature and certain extra regularity assumptions. We show that a closed $n$-manifold $M$ satisfying $\mathrm{diam}(M)^2\mathrm{sec}_M \geq -κ$ and $\mathrm{diam}(M)^2\mathrm{Ric}_M \geq -δ$, where $δ>0$ is sufficiently small depending only on $n$ and $κ$, fibers over a $b_1(M)$-torus. This removes the upper sectional curvature bound required in the earlier result of Yamaguchi \cite{Y88}. As a corollary, we obtain a refinement of Yamaguchi's smooth fibration theorem (\cite{Y91}), showing that the fiber itself (rather than a finite cover of it) fibers over a $b_1$-torus. Our results extend to manifolds satisfying a generalized Reifenberg condition introduced in \cite{HH24}, which encompasses both a lower bound on sectional curvature and the local rewinding Reifenberg condition. In the nonsmooth setting, a similar result also holds for a non-collapsed $\mathrm{RCD}(-ε(D,r,n),n)$ space whose diameter is bounded by $D$ and which satisfies the $(r,δ(n))$-local rewinding Reifenberg condition. The proofs rely on an equivariant regularity theorem for almost submetries under a lower Ricci curvature bound. In addition, we study the stability of rank of Abelian actions along equivariant Gromov-Hausdorff convergence in this paper. |
| title | Fibrations, the First Betti Number, and Almost Nonnegative Ricci Curvature |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2605.24380 |