发表机构
Département de mathématiques, Université de Fribourg; Mathematical Institute, University of Oxford; Centro de Investigación en Matemáticas（弗里堡大学; 牛津大学; 数学研究中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文利用局部整数流为度量流形及奇异空间建立可定向性理论，证明定向流边界定理，并揭示定向唯一性排除分支并蕴含Poincaré不等式。
AI 中文摘要
我们利用局部整数流发展了一套度量流形和奇异度量空间的可定向性理论。对于具有局部有限Hausdorff测度的拓扑$n$-流形，我们引入了局部度和测度论条件，在这些条件下，拓扑定向与可求积部分上边界为零、局部整数可求积且重数为1的$n$-流典范地对应。我们将这些流延拓到一类流形部分稠密且具有全测度的奇异空间，并在定向局部唯一性下建立了常值性和最高维同调结果。对于可定向的非塌缩$\rm{RCD}(\kappa,n)$空间，我们证明了流论意义上的Stokes定理：定向流的边界是几何边界的诱导定向流，且重数为1。特别地，其质量测度恰好是该边界上的$(n-1)$维Hausdorff测度。对于纯$n$维、局部测地完备且具有局部上曲率界的空间，我们证明了定向的局部唯一性排除了余维为一的分支，并蕴含局部$1$-Poincaré不等式。后一结果适用于同调流形之外，并将定量拓扑与Poincaré不等式之间的联系推广到此类奇异空间。
英文摘要
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.
