Twisted K-theory and obstructions against positive scalar curvature metrics

TL;DR

This paper investigates the twisted index obstruction in the context of positive scalar curvature metrics, demonstrating non-vanishing results for certain enlargeable spin manifolds through K-theoretic pairings.

Contribution

It introduces a decomposition of the twisted index obstruction and proves its non-vanishing for orientable enlargeable spin manifolds.

Findings

Non-vanishing of the twisted index obstruction for enlargeable spin manifolds

Decomposition of the twisted index into a pairing of twisted K-homology and K-theory

Implications for obstructions to positive scalar curvature

Abstract

We decompose the twisted index obstruction $\theta(M)$ against positive scalar curvature metrics for oriented manifolds with spin universal cover into a pairing of a twisted $K$-homology with a twisted $K$-theory class and prove that $\theta(M)$ does not vanish if $M$ is an orientable enlargeable manifold with spin universal cover.