Sweepouts of closed Riemannian manifolds

TL;DR

This paper proves the existence of controlled sweepouts of closed Riemannian manifolds by 1-cycles, relates the polyhedral waist to the filling radius, and explores bounds and generalizations involving volume, diameter, and higher dimensions.

Contribution

It introduces a new approach to sweepouts using polyhedral parameters, relates waist to filling radius, and extends results to higher-dimensional polyhedral sweepouts.

Findings

Existence of sweepouts with bounded length in terms of volume and diameter.

Polyhedral 1-dimensional waist equals the filling radius up to a constant.

Bounds for polyhedral waist of homology classes in terms of volume or diameter.

Abstract

We show that for every closed Riemannian manifold there exists a continuous family of $1$-cycles (defined as finite collections of disjoint closed curves) parametrized by a sphere and sweeping out the whole manifold so that the lengths of all connected closed curves are bounded in terms of the volume (or the diameter) and the dimension $n$ of the manifold, when $n \geq 3$. An alternative form of this result involves a modification of Gromov's definition of waist of sweepouts, where the space of parameters can be any finite polyhedron (and not necessarily a pseudomanifold). We demonstrate that the so-defined polyhedral $1$-dimensional waist of a closed Riemannian manifold is equal to its filling radius up to at most a constant factor. We also establish upper bounds for the polyhedral $1$-waist of some homology classes in terms of the volume or the diameter of the ambient manifold. In addition, we provide generalizations of these results for sweepouts by polyhedra of higher dimension using the homological filling functions. Finally, we demonstrate that the filling radius and the hypersphericity of a closed Riemannian manifold can be arbitrarily far apart.