Sweepouts of closed Riemannian manifolds (Q2232154)

In this paper that is rich of definitions and ideas, one main result is that \[ \frac{1}{(n+1)2^{n+1}}\mathrm{W}_1(M)\leq \mathrm{FillRad}(M)\leq \frac{1}{2}\mathrm{W}_1(M) \] for a closed Riemannian \(n\)-manifold \(M\). Here \(\mathrm{FillRad}(M)\) is the filling radius of \(M\). \(\mathrm{W}_1(M)\) is the homology \(1\)-waist (of \(M\)) defined by the authors and its definition will be given later. The result extends one of Sabourau's theorems in [the last author, Am. J. Math. 142, No. 4, 1051--1082 (2020; Zbl 1455.53069)] to the waist of multi-parameters families of one-cycle sweeping out any closed Riemannian manifold. A polyhedral \textit{\(p\)-sweepout} of \(M\) is a family \[ \varphi[h^{-1}(t)] \subseteq M \] with \(t \in T\), where \(h: N \to T\) is a continuous map from a closed \(n\)-pseudomanifold~\(N\) to a finite \((n-p)\)-dimensional polyhedron~\(T\) such that all fibers \(h^{-1}(t)\) are \(p\)-subpolyhedra of \(N\), and \(\varphi: N \to M\) is a continuous degree one map. That is, \[ \varphi_*([N]) = [M] \in H_n(M) \] where the homology coefficients are in~\(\mathbb{Z}\) if \(M\) is orientable, and in~\(\mathbb{Z}_2\) otherwise. Define the \textit{ homology \(p\)-waist} of a closed Riemannian manifold~\(M\) as \[ \mathrm{W}_p(M) = \inf_{\varphi,h}\sup_{t \in T} \mathrm{Vol}_p(\varphi_{|h^{-1}(t)}) \] where the infimum is taken over all \(n\)-pseudomanifolds~\(N\), all simplicial \((n-p)\)-complexes~\(T\), and all maps \(\varphi:N \to M\) and \(h:N \to T\) defining a polyhedral \(p\)-sweepout of~\(M\). Here, the notation \( \mathrm{Vol}_p(\varphi_{|h^{-1}(t)})\) stands for the volume of the map~\(\varphi\) restricted to the fiber~\(h^{-1}(t) \subseteq N\), and not merely the volume of its image (which might be smaller). If such sweepouts do not exist, the authors let \(\mathrm{W}_p(M)=0\). The differences between the definitions and \textit{M. Gromov}'s [Ann. Math. Stud. 205, 141--205 (2020; Zbl 1456.53035)] are that the authors do not require that \(T\) is a pseudomanifold. The authors offer an example to illustrate the distinction. Two other versions of homology 1-waist are defined and similar theorems are given in the paper. The authors also define \(\Phi\)-homotopy (\(p\), \(k\))-waist and give sweepout estimates for \(\Phi\)-homotopy (\(1\), \(k\))-waist. The authors construct a sequence (\(g_i\)) of Riemannian metrics on \(S^4\) with arbitrarily small hypersphericity and filling radius bounded away from zero. In the last part of the paper, they define the \(k\)-homological filling function and give the inequalities between \(\mathrm{W}_p(M)\) and \(k\)-homological filling functions with the diameter and the volume of \(M\).