On the Twisted KK-Theory and Positive Scalar Curvature Problem

TL;DR

This paper extends the understanding of obstructions to positive scalar curvature on manifolds by employing groupoid methods in KK-theory, removing previous restrictions on manifold dimension and spin structure assumptions.

Contribution

It introduces a new groupoid-based approach to prove the non-vanishing of the obstruction in more general cases, beyond even-dimensional or spin-structured manifolds.

Findings

Obstruction $ heta(M)$ does not vanish for enlargeable manifolds.

Groupoid methods generalize previous results to all dimensions.

The approach differs from prior methods by Hanke and Schick.

Abstract

Positiveness of scalar curvature and Ricci curvature requires vanishing the obstruction $\theta(M)$ which is computed in some KK-theory of C*-algebras index as a pairing of spin Dirac operator and Mishchenko bundle associated to the manifold. U. Pennig had proved that the obstruction $\theta(M)$ does not vanish if $M$ is an enlargeable closed oriented smooth manifold of even dimension larger than or equals to 3, the universal cover of which admits a spin structure. Using the equivariant cohomology of holonomy groupoids we prove the theorem in the general case without restriction of evenness of dimension. Our groupoid method is different from the method used by B. Hanke and T. Schick in reduction to the case of even dimension.