Orientability of metric manifolds and metric spaces with curvature bounds
Abstract
We develop a theory of orientability for metric manifolds and singular metric spaces using locally integral currents. For topological n-manifolds with locally finite Hausdorff measure, we introduce local degree and measure-theoretic conditions under which topological orientations correspond canonically to boundaryless, locally integer rectifiable n-currents of multiplicity one on the rectifiable part. We extend these currents to a class of singular spaces whose manifold part is dense and of full measure, and establish constancy and top-dimensional homology results under local uniqueness of orientation. For orientable non-collapsed RCD(κ,n) spaces, we prove a current-theoretic Stokes theorem: the boundary of the orientation current is the induced orientation current of the geometric boundary, with multiplicity one. In particular, its mass measure is precisely the (n−1)-dimensional Hausdorff measure on that boundary. For purely n-dimensional, locally geodesically complete spaces with local upper curvature bounds, we show that local uniqueness of orientation excludes codimension-one branching and implies a local $1$-Poincaré inequality. The latter result applies beyond homology manifolds and extends the connection between quantitative topology and Poincaré inequalities to this class of singular spaces.
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BibTeX
@misc{cavallucci2026orientability,
title = {{Orientability of metric manifolds and metric spaces with curvature bounds}},
author = {Nicola Cavallucci and Denis Marti and Andrea Mondino and Raquel Perales},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03968},
eprint = {2610.03968},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03968},
}
RIS
TY - UNPB
TI - Orientability of metric manifolds and metric spaces with curvature bounds
AU - Nicola Cavallucci
AU - Denis Marti
AU - Andrea Mondino
AU - Raquel Perales
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03968
UR - https://arxiv.org/abs/2610.03968
AB - We develop a theory of orientability for metric manifolds and singular metric spaces using locally integral currents. For topological n-manifolds with locally finite Hausdorff measure, we introduce local degree and measure-theoretic conditions under which topological orientations correspond canonically to boundaryless, locally integer rectifiable n-currents of multiplicity one on the rectifiable part. We extend these currents to a class of singular spaces whose manifold part is dense and of full measure, and establish constancy and top-dimensional homology results under local uniqueness of orientation. For orientable non-collapsed RCD(κ,n) spaces, we prove a current-theoretic Stokes theorem: the boundary of the orientation current is the induced orientation current of the geometric boundary, with multiplicity one. In particular, its mass measure is precisely the (n−1)-dimensional Hausdorff measure on that boundary. For purely n-dimensional, locally geodesically complete spaces with local upper curvature bounds, we show that local uniqueness of orientation excludes codimension-one branching and implies a local $1$-Poincaré inequality. The latter result applies beyond homology manifolds and extends the connection between quantitative topology and Poincaré inequalities to this class of singular spaces.
ER -