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No Connectedness Lemma for Alexandrov spaces
Abstract
Since its introduction in 2003, Wilking's Connectedness Lemma has become a powerful tool in the study of manifolds with positive sectional curvature. In recent work of the authors, this has been extended to orbifolds. The purpose of this note is to exhibit counterexamples in the class of Alexandrov spaces for the Connectedness Lemma in both its ordinary and dual formulation. This answers a 2007 question of Wilking.
Related
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BibTeX
@misc{nienhaus2026connectedness,
title = {{No Connectedness Lemma for Alexandrov spaces}},
author = {Jan Nienhaus and Dennis Wulle},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37476},
eprint = {2610.37476},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37476},
}
RIS
TY - UNPB
TI - No Connectedness Lemma for Alexandrov spaces
AU - Jan Nienhaus
AU - Dennis Wulle
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37476
UR - https://arxiv.org/abs/2610.37476
AB - Since its introduction in 2003, Wilking's Connectedness Lemma has become a powerful tool in the study of manifolds with positive sectional curvature. In recent work of the authors, this has been extended to orbifolds. The purpose of this note is to exhibit counterexamples in the class of Alexandrov spaces for the Connectedness Lemma in both its ordinary and dual formulation. This answers a 2007 question of Wilking.
ER -