Characterization of metric spaces with a metric fundamental class

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.