Characterization of metric spaces with a metric fundamental class

TL;DR

This paper establishes the equivalence of three geometric conditions on finite-volume metric manifolds with finite Nagata dimension, generalizing previous results and linking fundamental classes, index bounds, and Gromov--Hausdorff approximations.

Contribution

It proves the equivalence of three conditions on metric manifolds with finite Nagata dimension, extending known results to higher dimensions and broader classes.

Findings

All three conditions are equivalent for metric manifolds with finite Nagata dimension.

Without finite Nagata dimension, condition (1) implies (2) and (3) implies (1).

Generalizes approximation results to higher-dimensional metric manifolds with LLC and finite Nagata dimension.

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.