Skeletal Homology
Abstract
"Skeletal homology" Knε(X) refers to the homology of the chain complex Snε(X) generated by skeletal (n,ε)-simplices, i.e. functions from the $0$-skeleton of the standard simplex into a metric space X, with image diameter less than ε>0. This homology was previously defined by Goldfarb, who showed that for finite metric spaces, it is isomorphic to the simplicial homology HnΔ(VRε(X)) of the VR complex. We prove an isomorphism for arbitrary metric spaces, and introduce new methods to understand homology at scale. We define an invariant metric on Snε(X), called the ultradiamond metric, that extends the uniform metric on skeletal simplices. With this metric we prove that "close cycles are homologous", which quickly leads to a host of stability results. We modify methods from singular homology to prove a strong generalization of Hausmann's Theorem, one of the two main justifications to use HnΔ(VRε(X)) as a proxy for homology in discrete metric spaces. The second justification is Latchev's Theorem, for which we also prove a strong generalization. We define a homomorphism ρε:Hn(X)→ Knε(X) induced by repeated barycentric subdivision and restriction, the image of which we call "real homology" Hnε(X) at scale. We argue that Hnε(X) better represents bona fide homology at scale than HnΔ(VRε(X)). To distinguish them, we define "phantom homology" to be Pnε(X)=Knε(X)/Hnε(X), and use the stability of Knε(X) to show that in collapse of Riemannian manifolds (e.g. the Berger Spheres), phantom homology can anticipate the abrupt drop in dimension that occurs in the limit.
Related
BibTeX
@misc{dey2026skeletal,
title = {{Skeletal Homology}},
author = {Ivy Dey and Conrad Plaut},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2607.16009},
eprint = {2607.16009},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.16009},
}
RIS
TY - UNPB
TI - Skeletal Homology
AU - Ivy Dey
AU - Conrad Plaut
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2607.16009
UR - https://arxiv.org/abs/2607.16009
AB - "Skeletal homology" K_n^ε(X) refers to the homology of the chain complex S_n^ε(X) generated by skeletal (n,ε)-simplices, i.e. functions from the $0$-skeleton of the standard simplex into a metric space X, with image diameter less than ε>0. This homology was previously defined by Goldfarb, who showed that for finite metric spaces, it is isomorphic to the simplicial homology H_n^Δ(VR_ε(X)) of the VR complex. We prove an isomorphism for arbitrary metric spaces, and introduce new methods to understand homology at scale. We define an invariant metric on S_n^ε(X), called the ultradiamond metric, that extends the uniform metric on skeletal simplices. With this metric we prove that "close cycles are homologous", which quickly leads to a host of stability results. We modify methods from singular homology to prove a strong generalization of Hausmann's Theorem, one of the two main justifications to use H_n^Δ(VR_ε(X)) as a proxy for homology in discrete metric spaces. The second justification is Latchev's Theorem, for which we also prove a strong generalization. We define a homomorphism ρ_ε:H_n(X)→ K_n^ε(X) induced by repeated barycentric subdivision and restriction, the image of which we call "real homology" H_n^ε(X) at scale. We argue that H_n^ε(X) better represents bona fide homology at scale than H_n^Δ(VR_ε(X)). To distinguish them, we define "phantom homology" to be P_n^ε(X)=K_n^ε(X)/H_n^ε(X), and use the stability of K_n^ε(X) to show that in collapse of Riemannian manifolds (e.g. the Berger Spheres), phantom homology can anticipate the abrupt drop in dimension that occurs in the limit.
ER -