On Gromov's conjecture for totally non-spin manifolds (Q2828058)

The Gromov conjecture states that for every closed Riemannian \(n\)-manifold \((M,g)\) with positive scalar curvature the macroscopic dimension of the universal covering \(\tilde{M}\) satisfies NEWLINE\[NEWLINE\dim_{mc} \tilde{M} \leq n-2 NEWLINE\]NEWLINE for the pull-back metric on \(\tilde{M}\), see [\textit{M. Gromov}, Prog. Math. 132, 1--213 (1996; Zbl 0945.53022)]. The proof for \(n=3\) is known and may be found in [the first author, Math. Phys. Anal. Geom. 6, No. 3, 291--299 (2003; Zbl 1029.57016)]. So far the conjecture for the case of \(n\geq 5\) has been proved for some special cases of almost spin manifolds. The main goal of the presented paper is to show that the Gromov conjecture holds for totally non-spin manifolds with the exception when the virtual cohomogonical dimension of the fundamental group is not equal to the dimension of the manifold. At the same time let us recall that the solution for \(n=4\) is still unknown.