Maximal first Betti number drop and collapsing RCD spaces

1.Introduction

The interplay between the first Betti number and the topology of a closed Riemannian manifold with Ricci curvature bounded below has a rich history. The number is bounded above by the curvature and the dimension: Bochner’s theorem gives $b_1\le n$ when $\mathrm {Ric}\ge 0$ [Boc46], and Gromov’s short-generator argument gives the same bound under the pinching condition $\diam (M)^2\,\mathrm {Ric}_M\ge -\delta (n)$ [Gro81]. Attaining the bound brings rigidity. Equality in Bochner’s theorem forces a flat torus [Boc46], and Gromov’s conjecture that $b_1=n$ under the pinching condition forces a manifold diffeomorphic to $\T ^n$ was proved by Colding [Col97] up to homeomorphism and by Cheeger–Colding [CC97, App. A] up to diffeomorphism. See [HRW20] for a survey of these themes.

For collapsing sequences the same pattern appears, with the topology of the limit subtracted. Huang and Wang proved that if closed Riemannian $N$-manifolds $\{X_i\}$ with $\mathrm {Ric}\ge -(N-1)$ converge in the Gromov–Hausdorff sense to a closed $m$-manifold $X$ of bounded geometry, $m<N$, then

and that at equality the $X_i$ are torus bundles over $X$ [HW26, Thm. 1.1], an analogue under lower Ricci bounds of Fukaya’s fibration theorem for bounded sectional curvature [Fuk87]. The bundles arising this way are themselves rigid: Peng–Wang–Wang showed that at the same Betti number equality a smooth torus bundle with affine structure group over a closed manifold is principal, and that finite covers of base and total space turn it into a product [PWW26, Thm. 1.1].

It is natural to ask whether the above interplay between the first Betti number and lower Ricci curvature bounds generalizes to rougher spaces. The canonical candidates are $\RCD (K,N)$ spaces: metric measure spaces with Ricci curvature bounded below by $K$ and dimension bounded above by $N$ in a synthetic sense. This class is stable under measured Gromov–Hausdorff convergence and therefore contains all Ricci limit spaces. Every $\RCD (K,N)$ space $(Z,d,\mathfrak m)$ has a rectifiable (or essential) dimension $n\le N$: at $\mathfrak m$-almost every point, the tangent cone is uniquely $\R ^n$ [BS20]. We call

the regular set of $Z$, and its points regular. This set is dense and has full measure, although it need not be open.

Some of the results above already generalize to $\RCD$ spaces. Under the pinching condition $\diam (Z)^2\,K\ge -\delta (N)$, a compact $\RCD (K,N)$ space $Z$ has $b_1(Z)\le N$, and at equality it is bi-Hölder homeomorphic to a flat $N$-torus [MMP22, ZZ26]. The collapsing bound (1) also persists: it holds whenever compact $\RCD (K,N)$ spaces $\{(X_i,d_i,\mathfrak m_i)\}$ converge in the measured Gromov–Hausdorff sense to a space $(X,d,\mathfrak m)$ of rectifiable dimension $m$. This is [SRZ23, Thm. 5]. The argument of [HHWZ26, Thm. 6.12] derives the same bound; its rank step is repaired in Remark 3.2.

This paper studies the equality case for $\RCD$ spaces. Parts (i)–(iv) below require no regularity assumption on $X$ and produce the limit-cover symmetry, the orbifold good set, and local Seifert fibrations. The global-fibration and virtual-product conclusions in part (v), which extend the corresponding results of [HW26, PWW26], require no bubbling and a smooth closed Riemannian orbifold base. As a special case of our main result, we confirm Conjecture 2.9 of [ZZ26] that at equality of (1) the $X_i$ are torus bundles over $X$ whenever $X$ is a closed Riemannian manifold; see Corollary 9.1.

1.1. Setting and main result. Our main object is the limit of the maximal abelian covers of the $X_i$; in the equality case, these covers do not collapse. Throughout the paper we assume:

• $(X_i,d_i,\mathfrak m_i)$ are compact $\RCD (K,N)$ spaces with $N\ge 2$ and $\diam X_i\le D$, converging in the measured Gromov–Hausdorff sense to $(X,d,\mathfrak m)$ of rectifiable dimension $m$, and

  $$b_1(X_i)-b_1(X)=N-m=:k\ge 1\quad \text {for all }i .$$

The case $N<2$ is elementary, while $k=0$ is the non-collapsed case; Corollary 9.1 treats both for manifold limits. Let $\hat X_i\to X_i$ be the maximal abelian covers, with deck groups $H_i=H_1(X_i;\Z )$. After passing to a subsequence, let $(Y,H)$ be the equivariant pointed measured Gromov–Hausdorff limit of $(\hat X_i,H_i)$. We write $P\colon Y\to X$ for the projection and $f_i\colon X_i\to X$ for the Gromov–Hausdorff approximations. By part (i) below, $Y$ has rectifiable dimension $N$. Thus $\mathcal R(Y)$ consists of the points at which every tangent cone is $\R ^N$. It is $H$-invariant but need not be open (Remark 9.4). We say that $Y$ has no bubbling, or that (R) holds, when $\mathcal R(Y)=Y$.

Finite isotropy in the limit is unavoidable, already for surfaces collapsing to an interval (Example 9.2); in the smooth setting, Ricci-flow smoothing likewise produces local infranil fibrations over controlled Riemannian orbifold bases [HW22, Thm. 1.4]. The fibrations we produce are therefore Seifert fibrations in the following sense, with the orbifold group of the base point acting on the torus fibre by translations.

Part (i) is proved in §3, parts (ii) and (iii) in §5.3, part (iv) in §7.5, and part (v) in §8. All limit-dependent objects—including $Y,H,\Gamma ,C,W,W',G,\Pi _i$, and $X_i'$—and every “for large $i$” in the construction refer to the chosen subsequence. The covers $W'$ and $X_i'$ also depend on the chosen complement $\Gamma _{\mathrm f}$. Conclusions involving only $X_i$ and $X$ pass to the full sequence by the usual sub-subsequence argument.

Both qualifications in the theorem are genuine. Orbifold points can support exceptional fibres, as in the Klein-bottle collapse of Example 9.2, while bubbling can make $G$ strictly smaller than $X$ (Remark 9.4).

1.2. Contribution and proof spine. We first say what is imported and what each import supplies, then what is new, and then how the four stages fit together. Known inputs. The following results are used as black boxes. The bound (1) for $\RCD$ spaces and the rank inequality that keeps the maximal abelian covers from collapsing come from [SRZ23, Thm. 5] and [HHWZ26, Thm. 1.9]; Remark 3.2 records a gap in the posted proof of the latter and repairs it. Equivariant regularization over a smooth Riemannian target [HHWZ26, Thm. 1.11] supplies the global $C$-equivariant map used in part (v); this paper proves that it is a Seifert fibration. The affine infranil fibration theorem under local bounded covering geometry [Wan24a, Thm. B] supplies the separate second projection; the Betti equality is used here to show that its fibre is a torus. Finally, part (v) imports the classification of [PWW26, Thm. 1.1] quoted above, in which the covers are normal, of the same index, and have abelian deck groups, together with [PWW26, App. A], which matches topological principal torus bundles over a closed smooth manifold with smooth ones. These two inputs need an affine torus bundle over a closed smooth manifold, with the Betti number equality between total space and base; that is exactly what §8 has to produce.

New mechanism. For a singular limit, none of the results above simultaneously provides a slice for the non-compact residual action and a fibration compatible with finite isotropy. The central theme of this paper is to convert an asymptotic symmetry on the limit cover into exact product coordinates on the approximating spaces. We construct two complementary coordinates. The transverse coordinate is an $H_0$-invariant harmonic map, equivariant under finite isotropy, whose local level sets are exactly the $H_0$-orbits. The orbit coordinate is exactly equivariant under the discrete collapsing group. Together they form

the local product chart from which we read off the Seifert fibration.

Proof spine. The argument has four stages.

1.

Symmetry from equality (§3). Equality produces a non-collapsed limit $Y$ of the maximal abelian covers and a free action $H_0\cong \R ^k$; a rank-and-homogeneity argument at a regular-point blow-up supplies the dimension chain.

2.

The transverse coordinate (§§4–5). Harmonic cocycles $u\circ \varphi _v-u$, a Cheng–Yau estimate, and a dyadic pointwise linearization produce the infinitesimal orbit map; the transformation theorem normalizes it at the metric scale. Harmonic replacement on a fine lattice quotient, followed by averaging, produces the invariant slice and the orbifold charts on $G$.

3.

The orbit coordinate (§§6–7). Equivariant transverse maps first yield local bundles. Compatible quotient maps $\Pi _i\colon H_i\to \Gamma$ split the deck groups. Coordinatewise rounding and a doubling estimate realize $\ker \Pi _i$, modulo torsion, as a lattice in $\R ^k$ without passing to a further subsequence; a partition-of-unity construction then produces $U_i$ with exact deck equivariance. Pairing the two coordinates gives the local Seifert chart, which also shows that $\ker \Pi _i\cong \Z ^k$.

4.

Globalization (§8). Under no bubbling the charts cover $X$. For a smooth orbifold limit, unfolding makes $W'$ a smooth manifold. The local charts identify the fibres of the equivariantly regularized map and yield the Betti equality on the cover. Bounded covering geometry supplies a separate affine infranil bundle; the equality forces a torus fibre, and affine reduction gives the finite product cover.

Section 2 fixes notation and records the analytic packages used throughout; a reader familiar with splitting maps may begin at §3 and return to §2.3 whenever a package is invoked.

2.Background and reusable tools

This section fixes notation and collects the facts about $\RCD$ spaces, isometric actions, and equivariant convergence that are used repeatedly, together with the local analytic packages on which the later constructions rest.

2.1. Setup. We assume $(\ast )$ throughout. In particular, $N=m+k$ is an integer. The maximal abelian cover $\hat X_i\to X_i$ exists because $X_i$ is semi-locally simply connected [Wan24b]. We equip $\hat X_i$ with the lifted measure, normalized to give mass $1$ to the unit ball about a base point $\hat p_i$. After passing to a subsequence, $(\hat X_i,\hat p_i,H_i)\to (Y,\hat p,H)$ in the equivariant pointed measured Gromov–Hausdorff (pmGH) sense. The notation $\dim Y$ always refers to the rectifiable dimension associated with the limit measure. The objects to keep apart. The argument moves between the covers, the limit, and the approximating spaces. The following names are fixed once and for all.

• Upstairs: the maximal abelian covers $\hat X_i\to X_i$ with deck groups $H_i$, and their equivariant limit $(Y,H)$, which does not collapse.
• Limit quotients: the identity component $H_0\cong \R ^k$ of $H$; the transverse quotient $W=Y/H_0$; the intermediate quotient $W'=W/\Gamma _{\mathrm f}$; and $X=W'/C$ with $C$ finite abelian.
• Downstairs: the rank-$k$ fibre groups $H_i'\le H_i$, the finite covers $X_i'=\hat X_i/K_i$ on which $C$ acts freely, and the subgroups $\Lambda _i\le H_i$ stabilizing a small tube over a point of $X$ (§6).
• Coordinates: the $H_0$-invariant harmonic transverse map $\hat \psi$ and its approximants $\Phi _i$ (§§5 and 7.1), the exactly $\Lambda _i$-equivariant orbit map $U_i$, and the product chart $\mathcal Z_i=(U_i,\hat \Phi _i)$ (§7.4).
• The good set: the open, full-measure set $G\subset X$ over which the local Seifert fibrations are constructed (§5.3).

The covers and their limits fit into

where the horizontal arrows are quotient maps, the vertical ones denote equivariant measured Gromov–Hausdorff convergence as $i\to \infty$, and the transverse quotient $W=Y/H_0$ sits between $Y$ and $W'=W/\Gamma _{\mathrm f}$. The groups $K_i$ and $K$, and the covers $X_i'$, are constructed in §6.

Conventions. The symbol $\Psi (\epsilon _1,\dots ,\epsilon _l\,|\,c_1,\dots ,c_j)$ denotes a non-negative function tending to $0$ with $\epsilon _1,\dots ,\epsilon _l$, while $c_1,\dots ,c_j$ remain fixed. Its value may change from line to line, and arguments are suppressed when clear. Unless stated otherwise, scalar occurrences of $C,c>0$ denote constants depending only on $N$; when used as a group, $C$ always denotes the finite factor in Theorem 1.2(i). An $\epsilon$-GH approximation has distortion at most $\epsilon$ and $\epsilon$-dense image. A ball $B(z,s)$ is $\epsilon s$-close to $B^n(s)$ when $(B(z,s),s^{-1}d)$ is $\epsilon$-GH close to $B^n(1)$; measured closeness also compares the measures after normalizing both balls to mass $1$. A point is $n$-regular if all its tangent cones are $\R ^n$. Following [HHWZ26, Def. 2.4], rescaled to radius $r$, a map $u\colon B(x,r)\to \R ^j$ in the domain of the local Laplacian is a $(j,\delta )$-splitting map if

We omit $j$ when it is clear. We call $u$ harmonic when $\Delta u=0$. Our order of the parameters $j$ and $\delta$ is the reverse of that in [HHWZ26].

2.2. RCD spaces. We use the following facts about an $\RCD (K,N)$ space $(Z,d,\mathfrak m)$ with $N\in (1,\infty )$.

(P1)

Rectifiable dimension. We write $\dim Z$ for the rectifiable dimension $n$ of $Z$ (see the Introduction), which exists by [BS20]. If $n=N$, then $N\in \mathbb N$ and $\mathfrak m=c\,\mathcal H^N$ [BGHZ23]. Under pmGH convergence, $\dim$ is lower semicontinuous [Kit19, Thm. 1.5].

(P2)

Rescaling and non-collapsed spaces. For $\lambda ,c>0$, $(Z,\lambda d,c\,\mathfrak m)$ is $\RCD (\lambda ^{-2}K,N)$, because Ricci curvature scales as the inverse square of distance. The $\RCD$ condition is stable under pmGH convergence [GMS15], and being $\RCD (-\varepsilon ,N)$ for every $\varepsilon >0$ implies $\RCD (0,N)$. Consequently, pmGH limits of normalized rescalings $(Z,s_j^{-1}d,z_j)$ with $s_j\to 0$ are $\RCD (0,N)$ spaces, since the lower bounds $s_j^2K$ tend to $0$. In particular, every tangent cone is $\RCD (0,N)$. An $\RCD (K,N)$ space of the form $(Z,d,\mathcal H^N)$ is called non-collapsed [DPG18]. Its tangent cones are metric cones with measure $\mathcal H^N$, and $\mathcal H^N$ is continuous under GH convergence within this class (volume convergence).

(P3)

Isometries and covers. $\Isom (Z)$ is a Lie group [Sos18, GSR19]. $Z$ is semi-locally simply connected, in the local form: for every $z$ and $R>0$ there is $r>0$ such that every loop in $B_r(z)$ is contractible in $B_R(z)$ [Wan24b]. So every open ball has a universal cover. Covering spaces of $Z$, and quotients of $Z$ by free, properly discontinuous groups of measure-preserving isometries, are $\RCD (K,N)$, with the lifted or quotient measure. Indeed, the projection is a local isomorphism of metric measure spaces; $\RCD (K,N)$ spaces are non-branching [Den25], so the curvature-dimension condition globalizes [CM21, Li24].

(P4)

No small subgroups [ZZ26, Cor. 3.10]. Let $(X_i,d_i,\mathfrak m_i,p_i)$ be pointed $\RCD (K,N)$ spaces converging in the pmGH sense to a space of rectifiable dimension $N$. Then every sequence of subgroups $W_i\le \Isom (X_i)$ with $\sup _{g\in W_i,\,x\in B_R(p_i)}d(gx,x)\to 0$ for every $R$ is eventually trivial.

These are standard consequences of the Arzelà–Ascoli theorem.

Metric closeness to $\R ^m$ on a sufficiently large ball forces measured closeness on smaller balls. By contrast, closeness on only one fixed ball is insufficient: $(x_1+100)^{N-m}\mathcal L^m$ is an $\RCD (0,N)$ measure on $B^m(40)$.

2.3. Reusable local analytic tools. Several local analytic mechanisms recur throughout the proof. We record them here so that later arguments can focus on equivariance and topology rather than repeat the same estimates. Throughout this subsection, the displayed numerical radii may be replaced by any fixed nested radii; the constants and final inner radius then change only with that choice.

Volume to Reifenberg. Suppose a non-collapsed $\RCD (K,N)$ ball satisfies $|K|R^2=o(1)$ and has Bishop–Gromov ratio $1-o(1)$ at radius $R$. Monotonicity propagates this lower bound to every smaller radius. Volume almost rigidity [DPG18, Thm. 1.6], followed by volume convergence, then shows that the concentric balls of radius at most $R/2$ are measured-GH close to Euclidean balls. If the same control holds at every centre and scale in a region, the metric Reifenberg theorem [CC97, Thm. A.1.1] supplies the local topological-manifold structure, while canonical Reifenberg [HHWZ26, Thm. 2.10] gives the bi-Hölder estimates for the canonical coordinate maps. We refer to this chain of implications as the volume–Reifenberg package. In later applications, the only nonstandard step is to obtain the volume-ratio bound uniformly over the required family of centres.

The next lemma combines harmonic replacement with the comparison estimate used after averaging.

The first consequence makes approximate equivariance exact without losing the splitting estimates.

The second consequence allows prescribed coordinates to replace the corresponding coordinates of a full splitting chart.

The next elementary criterion packages the open-and-closed argument that turns a full Reifenberg chart with invariant transverse coordinates into an exact local orbit slice.

The following topological promotion will be used for both the local and the global transverse maps.