Vanishing Normalized Homology Under Local Manifold Convergence

Definitions and mathematical statement

Let $d\in{\mathbb{Z}}$ with $d\geq3$, and let $(M_n)_{n\geq1}$ be triangulated compact $d$-manifolds, with triangulations $K_n$. Write $V(K_n)$ for the vertex set and $K_n^{(1)}$ for the graph consisting of vertices and edges. The uniform degree assumption is that there exists $D<\infty$ such that \[ \deg_{K_n^{(1)}}(v)\leq D \quad\text{for every }n\geq1\text{ and every }v\in V(K_n). \] The source asks whether Benjamini--Schramm convergence to a triangulation of ${\mathbb{R}}^d$ implies \[ \lim_{n\to\infty}\frac{\dim H_1(M_n)}{|M_n|}=0. \] Here the coefficient system for homology, the meaning of $|M_n|$, the treatment of manifold boundary, and the precise rooted interpretation of the limit are not specified in the supplied record. Consequently, this displayed question does not yet determine a unique mathematical assertion.

One explicit conditional formulation, if finite triangulations, vertex normalization, homology over a fixed field, and simplicial Benjamini--Schramm convergence are intended, is as follows. Fix a field $\Bbbk$ and put \[ b_1(M_n;\Bbbk)=\dim_{\Bbbk}H_1(M_n;\Bbbk). \] Choose a root $o_n$ uniformly from the nonempty finite set $V(K_n)$. For a rooted simplicial complex $(K,o)$, let $B_r(K,o)$ be the rooted subcomplex induced by the vertices at graph distance at most $r$ from $o$ in $K^{(1)}$. Simplicial Benjamini--Schramm convergence to a random rooted locally finite complex $(T,o)$ means that, for every integer $r\geq0$ and every finite rooted simplicial complex $(A,a)$, \[ \lim_{n\to\infty}\Pr\left(B_r(K_n,o_n)\cong(A,a)\right) =\Pr\left(B_r(T,o)\cong(A,a)\right), \] where isomorphisms preserve the root and all simplices. Under the interpretation that $T$ triangulates ${\mathbb{R}}^d$ almost surely, the conditional question is whether every such sequence satisfies \[ \lim_{n\to\infty}\frac{b_1(M_n;\Bbbk)}{|V(K_n)|}=0. \] This conditional formulation supplies possible meanings for missing notation; it is not asserted to be the uniquely intended source formulation. In particular, neither the choice of $\Bbbk$ nor universal quantification over fields is supplied. The source also does not specify whether its limiting triangulation is fixed or random, whether convergence retains all simplices or only the $1$-skeleton, or whether compact manifolds with boundary are admitted. These qualifications remain unresolved; no connectedness or orientability hypothesis is supplied.

Short English statement

The question asks whether compact triangulated manifolds of dimension at least three, with uniformly bounded vertex degrees and local limit a triangulation of ${\mathbb{R}}^d$, must have vanishing normalized first homology. A precise assertion requires clarification of the homology coefficients, normalization, boundary convention, and local convergence model. The supplied status is partially solved; this draft does not independently verify that assessment.

Sources

• Local UnsolvedMath registry: supplied complete record for permanent ID $30004394$, titled Vanishing Normalized Homology Under Local Manifold Convergence, in the Oberwolfach Reports Open Problems collection. Exact source problem code: catalogue entry. The proposed year is recorded as 2020; no proposer is identified.
• Supplied source citation: Manifolds and Groups (2021), Oberwolfach Reports, DOI 10.4171/owr/2020/7, pp. 417--463. https://doi.org/10.4171/owr/2020/7. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Hutchcroft--Pete, Kazhdan groups have cost 1. https://arxiv.org/abs/1810.11015. The record does not identify a particular theorem establishing the stated manifold consequence.
• Supplied dated literature curation, checked at 2026-08-22: status partially solved. The supplied assessment says that the Benjamini--Schramm first-homology limit is known in dimension three, for covering towers in special settings, and under a Kazhdan-property hypothesis, while the general question remains open and includes the degree-one Singer problem. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The supplied background expressly requests independent verification of time-sensitive status and cited claims.