On Gromov's conjecture for totally non-spin manifolds

Abstract

Gromov's Conjecture states that for a closed $n$-manifold $M$ with positive scalar curvature the macroscopic dimension of its universal covering $\tilde{M}$ satisfies the inequality ${\dim }_{mc}\tilde{M}\le n-2$\cite{G2}. We prove this inequality for totally non-spin $n$-manifolds whose fundamental group is a virtual duality group with $vcd\ne n$. In the case of virtually abelian groups we reduce Gromov's Conjecture for totally non-spin manifolds to the vanishing problem whether $H_n(T^n{)}^{+}=0$ for the $n$-torus $T^n$ where $H_n(T^n{)}^{+}\subset H_n(T^n)$ is the subgroup of homology classes which can be realized by manifolds with positive scalar curvature.