A topological version of the Cartan-Hadamard Theorem and asphericity complexity
Abstract
We investigate a topological version of the Cartan-Hadamard theorem that allows one to study asphericity of spaces via distinguished families of paths. A topological space has the distinguished path property (dpp) if there exists a continuous map from the space Π(X) of homotopy classes of paths (relative endpoints) to the path space XI that is a right inverse to the canonical quotient map. For complete metric spaces endowed with a locally convex metric, the distinguished paths are precisely the local geodesics. We show that if X has the dpp, then its universal cover is contractible and, in particular, X is aspherical. The space Π(X) is a fiber bundle over X whose fiber is the universal cover of X, and in the case of Riemannian manifolds of non-positive curvature it is naturally isomorphic to the tangent bundle. We prove that every aspherical CW-complex that is locally finite or countable has the dpp, and this allows us to reinterpret asphericity of CW-complexes in terms of the existence of continuous sections. We also define and study the notion of asphericity complexity of spaces by means of local sections from Π(X) to XI and relate it to the concept of equivariant topological complexity introduced by Colman and Grant.
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BibTeX
@misc{minian2026topological,
title = {{A topological version of the Cartan-Hadamard Theorem and asphericity complexity}},
author = {Elias Gabriel Minian and Juan Martín Perez Garber},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04440},
eprint = {2609.04440},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04440},
}
RIS
TY - UNPB
TI - A topological version of the Cartan-Hadamard Theorem and asphericity complexity
AU - Elias Gabriel Minian
AU - Juan Martín Perez Garber
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.04440
UR - https://arxiv.org/abs/2609.04440
AB - We investigate a topological version of the Cartan-Hadamard theorem that allows one to study asphericity of spaces via distinguished families of paths. A topological space has the distinguished path property (dpp) if there exists a continuous map from the space Π(X) of homotopy classes of paths (relative endpoints) to the path space X^I that is a right inverse to the canonical quotient map. For complete metric spaces endowed with a locally convex metric, the distinguished paths are precisely the local geodesics. We show that if X has the dpp, then its universal cover is contractible and, in particular, X is aspherical. The space Π(X) is a fiber bundle over X whose fiber is the universal cover of X, and in the case of Riemannian manifolds of non-positive curvature it is naturally isomorphic to the tangent bundle. We prove that every aspherical CW-complex that is locally finite or countable has the dpp, and this allows us to reinterpret asphericity of CW-complexes in terms of the existence of continuous sections. We also define and study the notion of asphericity complexity of spaces by means of local sections from Π(X) to X^I and relate it to the concept of equivariant topological complexity introduced by Colman and Grant.
ER -