Macroscopic band width inequalities (Q2133962)

Estimating the width of Riemannian bands with positive scalar curvature has attracted a lot of attention since Gromov established band width inequalities for \(\mathbb{T}^n\times[0,1]\) with \(n\leq 8\) [\textit{M. Gromov}, Geom. Funct. Anal. 28, No. 3, 645--726 (2018; Zbl 1396.53068)]. For example, using the index theoretical approach, \textit{S. Cecchini} [Geom. Funct. Anal. 30, No. 5, 1183--1223 (2020; Zbl 1455.58008)] and \textit{R. Zeidler} [``Band width estimates via the Dirac operator'', Preprint, \url{arXiv:1905.08520}] proved band width inequalities for some spin manifolds. Following the trend, the author establishes band width inequalities for a new class of orientable smooth manifolds by applying a version of \textit{P. Papasoglu}'s theorem [Geom. Funct. Anal. 30, No. 2, 574--587 (2020; Zbl 1453.53050)] Papasoglu's theorem implies that if (\(X^n, g\)) is a proper Riemannian polyhedron and \(R > 0\) is a radius such that the volume of the \(R\)-ball \(B_R(x)\) is bounded from above by \(\epsilon_n R^n\) for every \(x\in X\), then the Uryson width \(UR_{n-1}(X, g)\leq R\), where \(\epsilon_n\) is a positive constant. Based on the theorem, the author works on macroscopic scalar curvature, defines filling-enlargeable manifolds \(M^{n-1}\) and shows that if \(g\) is a Riemannian metric on \(V:=M^{n-1}\times [0,1]\) with the property that all unit balls in the universal cover have volume less than \(\epsilon_n/2\), then width\((V,g)\leq 1\). \textit{M. Gromov} and \textit{H. B. Lawson jun.} defined an enlargeable smooth manifold in [Ann. Math. (2) 111, 209--230 (1980; Zbl 0445.53025)] and the reviewer defined enlargeable length-structures on closed topological manifolds in [Ann. Global Anal. Geom. 60, No. 2, 217--230 (2021; Zbl 1469.53119)]. Then the author calls a closed orientable manifold \(M^n\) filling-enlargeable if for every Riemannian metric \(g\) on \(M^n\) and every \(r > 0\), there is a Riemannian covering \(M^n_r\) of \(M^n\) with the filling radius \(\mathrm{FillRad}(M^n_r, g_r)\geq r\), where \(g_r\) denotes the lifted metric. The author also defines width-enlargeable by replacing the filling radius by Uryson width in the definition of filling-enlargeable manifolds. The author shows that manifolds has the following relationships: aspherical, enlargeable \(\subseteq\) filling-enlargeable \(\subseteq\) width-enlargeable \(\subseteq\) essential. In the last part of the paper, the author generalizes \textit{M. Brunnbauer} and \textit{B. Hanke}'s works [J. Topol. 3, No. 2, 463--486 (2010; Zbl 1196.53028)] to show some functorial properties of filling-enlargeable manifolds.