Simplicial volume and 0-strata of separating filtrations

TL;DR

This paper provides an alternative proof with explicit constants for a theorem relating the simplicial volume of a manifold to its Riemannian volume and local volume bounds, using area-minimizing separating sets.

Contribution

It introduces a new proof method for the Guth and Braun--Sauer theorem, utilizing Papasoglu's area-minimizing separating sets approach with explicit constants.

Findings

Established an explicit constant bound for simplicial volume in terms of volume and local volume bounds.

Provided an alternative proof technique for a known theorem, enhancing clarity and potential applications.

Connected geometric measure theory with topological invariants in Riemannian geometry.

Abstract

We use Papasoglu's method of area-minimizing separating sets to give an alternative proof, and explicit constants, for the following theorem of Guth and Braun--Sauer: If $M$ is a closed, oriented, $n$-dimensional manifold, with a Riemannian metric such that every ball of radius $1$ in the universal cover of $M$ has volume at most $V_1$, then the simplicial volume of $M$ is at most the volume of $M$ times a constant depending on $n$ and $V_1$.