The relative L^2 index theorem for Galois coverings

TL;DR

This paper establishes an L^2 index theorem for Galois coverings of spin manifolds with positive scalar curvature near infinity, linking spectral projections, higher indices, and classical index theorems.

Contribution

It introduces a new L^2 index for Galois coverings, compatible with higher index theory, and extends classical Gromov-Lawson index results to the L^2 setting.

Findings

Finite trace of spectral projections in the von Neumann algebra

Compatibility of L^2 index with Xie-Yu higher index

L^2 versions of Gromov-Lawson relative index theorems

Abstract

Given a Galois covering of complete spin manifolds where the base metric has PSC near infinity, we prove that for small enough epsilon > 0, the epsilon spectral projection of the Dirac operator has finite trace in the Atiyah von Neumann algebra. This allows us to define the L2 index in the even case and we prove its compatibility with the Xie-Yu higher index. We also deduce L2 versions of the classical Gromov-Lawson relative index theorems. Finally, we briefly discuss some Gromov-Lawson L2 invariants.