About the macroscopic dimension of certain PSC-manifolds
The notion of macroscopic dimension was introduced by \textit{M. Gromov} in [Functional analysis on the eve of the 21st century. Volume II. In honor of the eightieth birthday of I. M. Gelfand. Proceedings of a conference, held at Rutgers University, New Brunswick, NJ, USA, October 24-27, 1993. Boston, MA: Birkhäuser. Prog. Math. 132, 1--213 (1996; Zbl 0945.53022)]. A metric space is said to have \(\dim_\varepsilon V\leq k\) if there exists a \(k\)-dimensional polyhedron \(P\) and a proper uniformly co-bounded map \(\phi:V\to P\) with Diam\((\phi^{-1}(p))\leq\varepsilon\) for all \(p\in P\). Then, the macroscopic dimension of \(V\) is defined by \(\dim_{mc}V\leq k\) if \(\dim_\varepsilon V\leq k\) for some \(\varepsilon<\infty\). Gromov's Positive Scalar Curvature (PSC)-conjecture states: Let \((M^n,g)\) be a closed Riemannian PSC-manifold with torsion-free fundamental group and let \((\tilde{M^n},\tilde{g})\) be the universal covering of \(M^n\) with the pull-back metric. Then, \(\dim_{mc}(\tilde{M^n},\tilde{g})<n-1\). The main theorem of this paper proves a partial result in this direction. Namely, it is shown that if \(M^n\) is a closed spin PSC-manifold with cd\((\pi_1(M^n))\leq n-1\) and for which the strong Novikov conjecture holds for \(\pi_1(M^n\)); then Gromov's PSC-conjecture holds for \(M^n\).
scientific article; zbMATH DE number 5530571