On Gromov's scalar curvature conjecture

TL;DR

This paper proves Gromov's scalar curvature conjecture for certain spin manifolds with positive scalar curvature, assuming the Strong Novikov Conjecture and injectivity of a specific map related to KO-theory for their fundamental groups.

Contribution

It establishes the conjecture under new algebraic conditions on the fundamental group, linking geometric properties to algebraic K-theory and the Novikov Conjecture.

Findings

Proves Gromov's conjecture under specified algebraic conditions

Links scalar curvature properties to algebraic K-theory and Novikov Conjecture

Provides new insights into the topology of spin manifolds with positive scalar curvature

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: 1. The Strong Novikov Conjecture holds for $\pi$. 2. The natural map $per:ko_n(B\pi)\to KO_n(B\pi)$ is injective.