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Characterization of metric spaces with a metric fundamental class

Denis Marti, Elefterios Soultanis

TL;DR

The paper characterizes when a metric $n$-manifold of finite volume admits a metric fundamental class by linking three conditions: existence of a metric fundamental class, uniform local index bounds for Lipschitz maps, and Gromov--Hausdorff approximations by bi-Lipschitz manifolds with volume control. Under finite Nagata dimension, these conditions are equivalent, while without this assumption (1)$ ightarrow$(2) and (3)$ ightarrow$(1) still hold, yielding a robust framework connecting analytic, geometric, and topological features. The authors develop a perturbation/projection toolkit and a Nagata-dimension based approximation method to pass between metric currents and Lipschitz geometry, culminating in a construction that extends Ntalampekos–Romney’s surface result to higher dimensions with LLC. They further show that GH limits of LLC manifolds with volume control inherit a metric fundamental class, highlighting the persistence of topological invariants under metric convergence. These results advance the understanding of how metric, geometric, and topological structures interact in non-smooth settings, with potential implications for geometric group theory and analysis on metric spaces.

Abstract

We consider three conditions on metric manifolds with finite volume: (1) the existence of a metric fundamental class, (2) local index bounds for Lipschitz maps, and (3) Gromov--Hausdorff approximation with volume control by bi-Lipschitz manifolds. Condition (1) is known for metric manifolds satisfying the LLC condition by work of Basso--Marti--Wenger, while (3) is known for metric surfaces by work of Ntalampekos--Romney. We prove that for metric manifolds with finite Nagata dimension, all three conditions are equivalent and that without assuming finite Nagata dimension, (1) implies (2) and (3) implies (1). As a corollary we obtain a generalization of the approximation result of Ntalampekos--Romney to metric manifolds of dimension $n\ge 2$, which have the LLC property and finite Nagata dimension.

Characterization of metric spaces with a metric fundamental class

TL;DR

The paper characterizes when a metric -manifold of finite volume admits a metric fundamental class by linking three conditions: existence of a metric fundamental class, uniform local index bounds for Lipschitz maps, and Gromov--Hausdorff approximations by bi-Lipschitz manifolds with volume control. Under finite Nagata dimension, these conditions are equivalent, while without this assumption (1)(2) and (3)(1) still hold, yielding a robust framework connecting analytic, geometric, and topological features. The authors develop a perturbation/projection toolkit and a Nagata-dimension based approximation method to pass between metric currents and Lipschitz geometry, culminating in a construction that extends Ntalampekos–Romney’s surface result to higher dimensions with LLC. They further show that GH limits of LLC manifolds with volume control inherit a metric fundamental class, highlighting the persistence of topological invariants under metric convergence. These results advance the understanding of how metric, geometric, and topological structures interact in non-smooth settings, with potential implications for geometric group theory and analysis on metric spaces.

Abstract

We consider three conditions on metric manifolds with finite volume: (1) the existence of a metric fundamental class, (2) local index bounds for Lipschitz maps, and (3) Gromov--Hausdorff approximation with volume control by bi-Lipschitz manifolds. Condition (1) is known for metric manifolds satisfying the LLC condition by work of Basso--Marti--Wenger, while (3) is known for metric surfaces by work of Ntalampekos--Romney. We prove that for metric manifolds with finite Nagata dimension, all three conditions are equivalent and that without assuming finite Nagata dimension, (1) implies (2) and (3) implies (1). As a corollary we obtain a generalization of the approximation result of Ntalampekos--Romney to metric manifolds of dimension , which have the LLC property and finite Nagata dimension.

Paper Structure

This paper contains 18 sections, 25 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main result
  4. Outline
  5. Preliminaries
  6. Metric notions
  7. Rectifiability
  8. Simplicial complexes
  9. Nagata dimension
  10. Orientation and degree
  11. Metric currents
  12. Integer rectifiable and integral currents
  13. The homotopy formula
  14. Bounds for the local index
  15. Perturbations and projections
  16. ...and 3 more sections

Key Result

Theorem 1.2

Let $X$ be a metric $n$-manifold homeomorphic to a closed, oriented, connected smooth $n$-manifold $M$. Suppose $\mathcal{H}^n(X)<\infty$ and $\dim_NX<\infty$. Then the following properties are equivalent.

Theorems & Definitions (52)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • proof
  • Corollary 1.4
  • proof
  • Lemma 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Definition 2.4
  • ...and 42 more