Macroscopic dimension of 3-manifolds
The concept of macroscopic dimension was introduced by \textit{M.~Gromov} [Functional analysis on the eve of the 21st century, Vol. II (New Brunswick, NJ, 1993), Boston, MA: Birkhäuser. Prog. Math. 132, 1-213 (1996; Zbl 0945.53022)]. A metric space \(V\) has macroscopic dimension at most \(k\) (\(\text{dim}_{mc} V \leq k\)) if there is a \(k\)-dimensional polyhedron \(P\) and a proper uniformly co-bounded map \(\varphi : V \to P\) that is a proper map such that \(\text{Diam} (\varphi^{-1} (p)) \leq \varepsilon\) for all \(p \in P\) and some possibly large \(\varepsilon < \infty\). If \(k\) is as minimal as possible, we say that \(\text{dim}_{mc} V = k\). It was conjectured by M.~Gromov that if the universal isometric covering \(\widetilde{M}\) of a compact \(n\)-dimensional Riemannian manifold \(M\) has \(\text{dim}_{mc} \widetilde{M} < n\), then \(\text{dim}_{mc} \widetilde{M} < n-1\). The author gives a positive answer to Gromov's conjecture in the case \(n=3\): the macroscopic dimension of the universal covering \(\widetilde{M}\) of a closed Riemannian 3-manifold \(M\) cannot be equal to 2.
scientific article; zbMATH DE number 1985808