Positive Scalar Curvature due to the Cokernel of the Classifying Map

TL;DR

This paper investigates the classification of positive scalar curvature metrics on spin manifolds, linking the existence and diversity of such metrics to algebraic invariants and conjectures in topology.

Contribution

It establishes lower bounds on the number of positive scalar curvature metrics up to bordism based on the cokernel of the classifying map, assuming the rational analytic Novikov conjecture.

Findings

Provides bounds on psc metrics related to KO-theory maps

Connects geometric properties to algebraic invariants of fundamental groups

Assumes the rational analytic Novikov conjecture for results

Abstract

This paper contributes to the classification of positive scalar curvature metrics up to bordism and up to concordance. Let $M$ be a closed spin manifold of dimension $\ge 5$ which admits a metric with positive scalar curvature. We give lower bounds on the rank of the group of psc metrics over $M$ up to bordism in terms of the corank of the canonical map $KO_*(M)\to KO_*(B\pi_1(M))$, provided the rational analytic Novikov conjecture is true for $\pi_1(M)$.