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 \(n \ge 3\). 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.
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Notes
In the proof of Proposition 4.5, we relax the usual definition of a pseudomanifold to a finite disjoint union of pseudomanifolds allowing a pseudomanifold to be non-connected.
\(\mathcal {R}\) stands for “replacement".
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Acknowledgements
This research has been partially supported by NSERC Discovery Grants RGPIN-2017-06068 and RGPIN-2018-04523 of the first two authors. The third author would like to thank the Fields Institute and the Department of Mathematics at the University of Toronto for their hospitality where a large part of this work was done.
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Partially supported by the ANR project Min-Max (ANR-19-CE40-0014).
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Nabutovsky, A., Rotman, R. & Sabourau, S. Sweepouts of closed Riemannian manifolds. Geom. Funct. Anal. 31, 721–766 (2021). https://doi.org/10.1007/s00039-021-00575-3
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DOI: https://doi.org/10.1007/s00039-021-00575-3