ℤk-smooth approximation of mod k integral cycles
Abstract
We prove that every mod k integral cycle T of sufficiently high codimension in an oriented Riemannian manifold ℳ can be approximated in flat norm by a cycle which is a ℤk-smooth submanifold Σ of nearly the same area, up to a singular set of codimension 5. If in addition T belongs to a mod k homology class admitting an embedded ℤk-smooth manifold representative, then Σ can be taken fully regular, up to the codimension one stratum of classical singularities. This extends the smooth approximation theory to cycles with coefficients in ℤ/kℤ in stable regimes. Moreover, we provide examples showing the impossibility of locally approximating minimizing cycles by smooth or ℤk-smooth submanifolds, even in homology classes with smooth or ℤk-smooth representatives.
Related
BibTeX
@misc{caldini2026ℤksmooth,
title = {{$\mathbbZ_k$-smooth approximation of mod $k$ integral cycles}},
author = {Gianmarco Caldini and Anna Skorobogatova},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20499},
eprint = {2610.20499},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20499},
}
RIS
TY - UNPB
TI - ℤ_k-smooth approximation of mod k integral cycles
AU - Gianmarco Caldini
AU - Anna Skorobogatova
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20499
UR - https://arxiv.org/abs/2610.20499
AB - We prove that every mod k integral cycle T of sufficiently high codimension in an oriented Riemannian manifold ℳ can be approximated in flat norm by a cycle which is a ℤ_k-smooth submanifold Σ of nearly the same area, up to a singular set of codimension 5. If in addition T belongs to a mod k homology class admitting an embedded ℤ_k-smooth manifold representative, then Σ can be taken fully regular, up to the codimension one stratum of classical singularities. This extends the smooth approximation theory to cycles with coefficients in ℤ/kℤ in stable regimes. Moreover, we provide examples showing the impossibility of locally approximating minimizing cycles by smooth or ℤ_k-smooth submanifolds, even in homology classes with smooth or ℤ_k-smooth representatives.
ER -