On Gromov's positive scalar curvature conjecture for duality groups
Let \(X\) be a metric space. The \textit{macroscopic dimension} \(\dim_{mc}X\) of \(X\) is defined as the minimum \(k\) such that there is a continuous uniformly cobounded map \(f:X\rightarrow N^{k},\) where \(N^k\) is a \(k\)-dimensional simplicial complex, i.e. there is a constant \(C>0\) with \(\mathrm{diam}(f^{-1}(y))<C\) for any \(y\in N^{k}\). \textit{M. Gromov} in [Prog. Math. 132, 1--213 (1996; Zbl 0945.53022)] conjectured that \(\dim_{mc}\tilde{M}\leq n-2,\) the macroscopic dimension of the universal cover \(\tilde{M}\) of a closed \(n\)-manifold \(M\) with a positive scalar curvature metric is at most \(n-2\). In the article under review, the author gives a characterization of macroscopic dimension in terms of coarsely equivariant cohomology. It is proved that \(\dim_{mc}\tilde{M}\leq n-2\) for almost spin \(n\)-manifolds \(M\) with positive scalar curvature whose fundamental group \(\pi_{1}(M)\) is a virtually duality group satisfying the coarse Baum-Connes conjecture.
scientific article; zbMATH DE number 6327952
scientific article