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On open covers of aspherical spaces satisfying π1-constraints and bounded Adamson cohomology

  • Arturo Espinosa Baro
  • Stephan Mescher
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Abstract

Given a topological space X and a family of subgroups ℱ of π1(X), the ℱ-category of X is given as one less than the cardinality of the smallest cover of X by open subsets whose fundamental groups lie in ℱ. In this article we study ℱ-categories of aspherical spaces and obtain cohomological lower bounds, a maximality result and bounded cohomology classes whose vanishing properties are crucial for determining ℱ-categories. For this purpose, we study the bounded Adamson cohomology of a family of subgroups and discuss universality properties of bounded cohomology classes. We apply our techniques to derive some applications to the study of the monotonicity of ℱ-categories under degree-one maps between manifolds and to provide a counterexample to a question of Capovilla, Löh and Moraschini.

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Cite

BibTeXRIS
BibTeX
@misc{baro2026open,
  title = {{On open covers of aspherical spaces satisfying $π_1$-constraints and bounded Adamson cohomology}},
  author = {Arturo Espinosa Baro and Stephan Mescher},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.07932},
  eprint = {2610.07932},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.07932},
}
RIS
TY  - UNPB
TI  - On open covers of aspherical spaces satisfying π_1-constraints and bounded Adamson cohomology
AU  - Arturo Espinosa Baro
AU  - Stephan Mescher
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.07932
UR  - https://arxiv.org/abs/2610.07932
AB  - Given a topological space X and a family of subgroups ℱ of π_1(X), the ℱ-category of X is given as one less than the cardinality of the smallest cover of X by open subsets whose fundamental groups lie in ℱ. In this article we study ℱ-categories of aspherical spaces and obtain cohomological lower bounds, a maximality result and bounded cohomology classes whose vanishing properties are crucial for determining ℱ-categories. For this purpose, we study the bounded Adamson cohomology of a family of subgroups and discuss universality properties of bounded cohomology classes. We apply our techniques to derive some applications to the study of the monotonicity of ℱ-categories under degree-one maps between manifolds and to provide a counterexample to a question of Capovilla, Löh and Moraschini.
ER  -