On Gromov’s scalar curvature conjecture

Abstract

We prove the Gromov conjecture on the macroscopic dimension of the universal covering of a closed spin manifold with a positive scalar curvature under the following assumptions on the fundamental group.0.10.1.Theorem.Suppose that a discrete groupπ\pihas the following properties:11. The Strong Novikov Conjecture holds forπ\pi.22. The natural mapper:kon(Bπ)→KOn(Bπ)per:ko_n(B\pi )\to KO_n(B\pi )is injective. Then the Gromov Macroscopic Dimension Conjecture holds true for spinnn-manifoldsMMwith the fundamental groupsπ1(M)\pi _1(M)that containπ\pias a finite index subgroup.