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Examples of ℤ/2-Harmonic 1-Forms
Abstract
In this paper, we develop methods for constructing ℤ/2-harmonic 1-forms in dimension three by varying the background metric. On ℝ3, we construct an example for which the complement of the smooth locus in the singular set is a Cantor set. We realize every finite graph with positive even valence at each vertex as the monodromy locus of a ℤ/2-harmonic 1-form on B3. We also construct desingularization models for every critical ℤ/2-eigensection, agreeing exactly with the associated homogeneous model outside a compact set. Finally, we construct a nondegenerate ℤ/2-harmonic 1-form on every closed connected oriented three-manifold for a suitable smooth metric.
Related
BibTeX
@misc{chen2026examples,
title = {{Examples of $\mathbbZ/2$-Harmonic 1-Forms}},
author = {Jiahuang Chen and Siqi He},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18915},
eprint = {2609.18915},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18915},
}
RIS
TY - UNPB
TI - Examples of ℤ/2-Harmonic 1-Forms
AU - Jiahuang Chen
AU - Siqi He
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18915
UR - https://arxiv.org/abs/2609.18915
AB - In this paper, we develop methods for constructing ℤ/2-harmonic 1-forms in dimension three by varying the background metric. On ℝ^3, we construct an example for which the complement of the smooth locus in the singular set is a Cantor set. We realize every finite graph with positive even valence at each vertex as the monodromy locus of a ℤ/2-harmonic 1-form on B^3. We also construct desingularization models for every critical ℤ/2-eigensection, agreeing exactly with the associated homogeneous model outside a compact set. Finally, we construct a nondegenerate ℤ/2-harmonic 1-form on every closed connected oriented three-manifold for a suitable smooth metric.
ER -